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לא יהיה כלי גבר על אשה. מנע הכתוב כלי זיין מן האשה כדי שלא תצא למלחמה ויהיה סבה לזנות, וכן האיש אם ילבש שמלת אשה ושיתערב עם הנשים, כל זה תועבת יהוה. וכן דרשו רז"ל כי תועבת יהוה אלהיך, דבר המביא לידי תועבה. ומן הכתוב הזה למדנו רז"ל מפי השמועה שאסור לו לאדם להתקשט בקשוט המיוחד להן, כגון לראות במראה וכיוצא בו, והוא אחד מן השלשה דברים שהוצרכו חכמים להתיר לבית רבי, ושמע מינה שאסור הוא לשאר בני אדם שיסתכלו במראה לפי שהוא קשוטי הנשים אלא א"כ היה סופר שהתירו לו להסתכל כדי לחזק כח הראות, או שהוא חולה וחלוש ורוצה להסתכל בתאר פניו, הרי זה מותר, שאין זה בכלל קשוט ואינו דרך תאוה ופתויי יצר הרע. ועוד אמרו רז"ל לקט לבנות מתוך שחורות חייב משום לא ילבש גבר שמלת אשה. ועוד דרשו רבותינו ז"ל כי מה שהרגה יעל לסיסרא ביתד ולא הרגתו בסכין, משום לא יהיה כלי גבר על אשה.

לא יהיה כלי גבר על אשה, “male apparel shall not be on a woman.” The principal concern of our verse is to deny the woman the right to bear arms (Sifri 227, Nazir 59). She is not to go to war and become a cause for immorality rampant during war. Similar considerations, i.e. the prevention of creating opportunities for sexual promiscuity- are the reason males are forbidden to wear women’s garments. Both of these apparent role reversals of the sexes are an abomination to the Lord.
Our sages in Jerusalem Talmud Shabbat 6,1 understand this wording to mean that “garments which will result in, or lead to abominations being perpetrated are forbidden” (compare Yoreh Deyah 156,2). Jewelry which is associated specifically with women is forbidden to be worn by men, as are mannerisms practiced especially by women, such as looking in the mirror. Seeing that they were one of three things which the rabbis had to give a special dispensation to for the household of Rabbi Yehudah Hanassi, it is clear that generally speaking male Jews are not supposed to admire themselves in a mirror. If a barber, in order to trim the client’s hair, needs to look into the mirror to better see the hairs he is looking for, this is in order. If a person was sick and he wishes to reassure himself that he is on the way to recovery by consulting a mirror and confirming that he looks healthier, this too is in order. The last mentioned examples are not uses for the enhancement of one’s exterior, for “dolling oneself up.” Our sages in Shabbat 94 also stated that if one picks a single white hair out of the black hairs in one’s beard one is guilty of violating the basic prohibition not to wear women’s clothing. A further illustration of this subject is found in Yalkut Shimoni on Judges item 56, that the reason Yael slew Siserah with a nail instead of with a knife was that she did not want to violate the commandment of using implements reserved for males.

[...]This is reinforced by the new prohibitions detailed in this commentary. There are many prohibitions against men shaving or anything else that could be considered self-beautification, including the use of a mirror. Men are not supposed to be sexually desirable, according to many ancient rabbis. Unfortunately, men are very attractive unless they specifically guard against looking desirable.

I'm not intentionally reading any specific rabbi as homosexual, and I have no explanation for why this is so commonly repeated in commentaries, but the homoerotic undertones of these prohibitions are too obvious to ignore.

I made a joke about this strange prohibition recently ("fellas, does it trans your gender to see a mirror?") but wanted to quote the Talmudic source directly - and I ran into some odd hiccups.
The concept of looking in a mirror as inherently feminine and therefore prohibited is mentioned here and there in commentaries, but I have no clue where it came from. The answer may be in Hebrew, which I sadly cannot read. I have found mentions that previous rabbis enforced this prohibition, but no one actually claiming that it is a current prohibition.
I now have the impression that this anti-mirror rule is a misinterpretation of the Talmud that happened centuries after the Talmud itself, and an obscure one at that. I have no proof one way or the other, except that it is downright bizarre for the Talmud to leave a rule so fundamental to one's identity, manhood, authority, safety, and social perception be merely implied. If Rabbeinu Bahya misread the source text, the commentary begins to make a lot more sense.
Below is a more detailed discussion of the various citations I've found for the anti-mirror rule.
לֹא יֵצֵא הָאִישׁ לֹא בְסַיִף, וְלֹא בְקֶשֶׁת, וְלֹא בִתְרִיס, וְלֹא בְאַלָּה, וְלֹא בְרֹמַח. וְאִם יָצָא, חַיָּב חַטָּאת. רַבִּי אֱלִיעֶזֶר אוֹמֵר, תַּכְשִׁיטִין הֵן לוֹ. וַחֲכָמִים אוֹמְרִים, אֵינָן אֶלָּא לִגְנַאי, שֶׁנֶּאֱמַר (ישעיה ב) וְכִתְּתוּ חַרְבוֹתָם לְאִתִּים וַחֲנִיתוֹתֵיהֶם לְמַזְמֵרוֹת, לֹא יִשָּׂא גּוֹי אֶל גּוֹי חֶרֶב וְלֹא יִלְמְדוּ עוֹד מִלְחָמָה. בִּירִית, טְהוֹרָה, וְיוֹצְאִין בָּהּ בְּשַׁבָּת. כְּבָלִים, טְמֵאִין, וְאֵין יוֹצְאִין בָּהֶם בְּשַׁבָּת:

Just as it is prohibited for a woman to carry out certain items unique to a woman into the public domain, the Sages said that a man may neither go out on Shabbat with a sword, nor with a bow, nor with a shield [teris], nor with an alla, nor with a spear. And if he unwittingly went out with one of these weapons to the public domain he is liable to bring a sin-offering. Rabbi Eliezer says: These weapons are ornaments for him; just as a man is permitted to go out into the public domain with other ornaments, he is permitted to go out with weapons. And the Rabbis say: They are nothing other than reprehensible and in the future they will be eliminated, as it is written: “And they shall beat their swords into plowshares and their spears into pruning hooks; nation will not raise sword against nation, neither will they learn war anymore” (Isaiah 2:4). With regard to women’s ornaments, they added that a garter placed on her leg to hold up stockings is pure and cannot become ritually impure as a utensil, and she may even go out with it on Shabbat. However, ankle chains, which were also women’s ornaments, can become ritually impure, and she may not go out with them on Shabbat.

As the translator explains, this passage (along with the rest of Mishnah Shabbat 6, which is mostly about women's ornaments and mentions no mirrors) is specifically for people conforming to expected gender roles. It discusses "a woman [carrying] out certain items unique to a woman into the public domain" and men carrying items unique to men. There is no mention of gender transgression whatsoever.
I won't argue that this is empowering (technically, maybe men can wear girly jewelry on Shabbat? Another diversity win!), but it clearly does not address the topic of crossdressing or gender nonconformity in the first place.
The Mishnah doesn't even condemn jewelry, perfumes, or clothes as womanly vanity - the worst it has to say is about men carrying weapons in a civilian setting at all - but instead with what is allowed on Shabbat, hence the name.
מַתְנִי׳ הַנּוֹטֵל צִפׇּרְנָיו זוֹ בָּזוֹ, אוֹ בְּשִׁינָּיו, וְכֵן שְׂעָרוֹ, וְכֵן שְׂפָמוֹ, וְכֵן זְקָנוֹ, וְכֵן הַגּוֹדֶלֶת, וְכֵן הַכּוֹחֶלֶת, וְכֵן הַפּוֹקֶסֶת — רַבִּי אֱלִיעֶזֶר מְחַיֵּיב, וַחֲכָמִים אוֹסְרִין מִשּׁוּם שְׁבוּת.
MISHNA: With regard to one who removes his fingernails with one another on Shabbat without scissors, or with his teeth, and the same is true with regard to one who removes his hair with his hands, and the same is true with regard to his mustache, and the same is true with regard to his beard, and the same is true with regard to a woman who braids her hair, and the same is true with regard to one who applies blue eye shadow, and the same is true with regard to one who applies blush, Rabbi Eliezer deems them all liable, as they each performed a labor prohibited by Torah law. And the Rabbis prohibited performing all of these actions due to rabbinic decree. None of the actions constitute prohibited labors.
תַּנְיָא נָמֵי הָכִי: הַנּוֹטֵל מְלֹא פִי הַזּוּג בְּשַׁבָּת — חַיָּיב. וְכַמָּה מְלֹא פִי הַזּוּג — שְׁתַּיִם. רַבִּי אֱלִיעֶזֶר אוֹמֵר: אַחַת, וּמוֹדִים חֲכָמִים לְרַבִּי אֱלִיעֶזֶר בִּמְלַקֵּט לְבָנוֹת מִתּוֹךְ שְׁחוֹרוֹת, שֶׁאֲפִילּוּ אַחַת חַיָּיב. וְדָבָר זֶה אַף בַּחוֹל אָסוּר, מִשּׁוּם שֶׁנֶּאֱמַר: ״לֹא יִלְבַּשׁ גֶּבֶר שִׂמְלַת אִשָּׁה״.
That was also taught in a baraita: One who removes enough of his hair to fill the opening of the scissors on Shabbat is liable. And how much is enough to fill the opening of the scissors? It is two hairs. Rabbi Eliezer says: One is liable for removing even one hair. And the Sages agree with Rabbi Eliezer that one who collects and plucks white hairs from among black ones is liable even if he removed a single hair. His actions indicate that one hair is significant for him. And this matter of plucking white hairs is prohibited for men even on weekdays, as it is stated: “A woman shall not don a man’s clothes, and a man shall not wear a woman’s garment” (Deuteronomy 22:5). The Sages derive that any action typically performed by women for beautification is prohibited for men.
I might like to make the argument that the prohibition must be against shaving one's beard to pass as a woman, as mentioned in Nazir 58b:10 in the source sheet. Putting that aside and taking for granted that plucking hair is a womanly behavior because of beautification, mirrors are still mysteriously absent. They come up again later:
וְהָנֵי תַנָּאֵי כְּהָנֵי תַנָּאֵי. דְּתַנְיָא: אֵין רוֹאִין בְּמַרְאָה בְּשַׁבָּת. רַבִּי מֵאִיר מַתִּיר בְּמַרְאָה הַקְּבוּעָה בַּכּוֹתֶל. מַאי שְׁנָא הַקְּבוּעָה בַּכּוֹתֶל — דְּאַדְּהָכִי וְהָכִי מִדְּכַר, שֶׁאֵינוֹ קָבוּעַ נָמֵי: אַדְּהָכִי וְהָכִי מִדְּכַר! הָכָא בְּמַרְאָה שֶׁל מַתֶּכֶת עָסְקִינַן, וְכִדְרַב נַחְמָן אָמַר רַבָּה בַּר אֲבוּהּ. דְּאָמַר רַב נַחְמָן אָמַר רַבָּה בַּר אֲבוּהּ: מִפְּנֵי מָה אָמְרוּ מַרְאָה שֶׁל מַתֶּכֶת אֲסוּרָה — מִפְּנֵי שֶׁאָדָם עָשׂוּי לְהַשִּׁיר בָּהּ נִימִין הַמְדוּלְדָּלִין.

The Gemara comments that in this matter, these tanna’im are like those tanna’im, who also argued over the same principle, as it was taught in a baraita: One may not look in a mirror on Shabbat lest one see a hair hanging and pluck it. Rabbi Meir permits looking in a mirror that is fixed on a wall.

The Gemara questions Rabbi Meir’s leniency: What is different about a mirror that is fixed on a wall? In that situation we say that, in the meantime, while one goes to bring scissors or another appliance to cut one’s hair, one will remember that it is Shabbat and that it is prohibited to cut hair. If so, with regard to a mirror that is not fixed on a wall, we can also say that in the meantime one will remember.

Rather, here we are dealing with a metal mirror, and it is as Rav Naḥman said that Rabba bar Avuh said, for Rav Naḥman said that Rabba bar Avuh said: For what reason did the Sages say that a metal mirror is prohibited for use on Shabbat? Because a person may remove hanging hairs with it, meaning that one may use the sharp edge of the mirror itself to cut the hairs. If the mirror is permanently set on the wall, we are not concerned that one will do this. This is similar to the view that one may read writing that is high up on a wall because it is impractical to erase the writing.

The prohibition against mirrors is remarkably grounded, although it deals with a type of mirror that hardly matches our modern ones. These mirrors were held in the hand but with sharp enough edges to cut hair. (How realistic this expectation was, I can't tell you.) The rule is against using mirrors on Shabbat in a way that would encourage prohibited labor. The mirrors themselves are blameless, as are the men using them.
"What about barbers?" I can hear you asking. (No?) They were presumed barbaric until proven innocent, and even then it was a close shave:
ואין מסתפרין מהן בכל מקום: ת"ר ישראל המסתפר מעובד כוכבים רואה במראה ועובד כוכבים המסתפר מישראל כיון שהגיע לבלוריתו שומט את ידו אמר מר ישראל המסתפר מעובד כוכבים רואה במראה היכי דמי אי ברשות הרבים ל"ל מראה ואי ברשות היחיד כי רואה מאי הוי לעולם ברה"י וכיון דאיכא מראה מתחזי כאדם חשוב רב חנא בר ביזנא הוה מסתפר מעובד כוכבים בשבילי דנהרדעא א"ל חנא חנא יאי קועיך לזוגא אמר תיתי לי דעברי אדר"מ ואדרבנן לא עבר אימר דאמור רבנן ברה"ר ברה"י מי אמור והוא סבר שבילי דנהרדעא כיון דשכיחי רבים כרה"ר דמו: ועובד כוכבים המסתפר מישראל כיון שהגיע לבלוריתו שומט את ידו: וכמה אמר רב מלכיה אמר רב אדא בר אהבה שלשה אצבעות לכל רוח ורוח

§ The mishna teaches: And one may not have his hair cut by gentiles anywhere. The Sages taught in a baraita: A Jew who has his hair cut by a gentile should observe the gentile’s actions in a mirror while he cuts his hair. And in the case of a gentile who has his hair cut by a Jew, when the Jew reaches the gentile’s forelock he removes his hand and does not cut it, because it is associated with idol worship.

The Master said: A Jew who has his hair cut by a gentile should observe the gentile’s actions in a mirror. The Gemara asks: What are the circumstances of this case? If it is referring to a haircut performed in a public domain, why do I need a mirror? After all, the gentile will not harm a Jew in public. And if it occurs in a private domain, even if the Jews observes the gentile’s actions, what of it? How does the fact that the Jew is watching prevent the gentile barber from harming him? The Gemara explains: Actually, this is referring to a haircut in a private domain, but since there is a mirror in place, the Jew appears as an important person whom the gentile will hesitate to attack.

The Gemara relates a relevant incident: Rav Ḥana bar Bizna was having his hair cut by a gentile in one of the side streets of Neharde’a. The barber said to him: Ḥana, Ḥana; Your throat is appealing to the razor. Rav Ḥana bar Bizna said: I have this coming to me, as I violated the ruling of Rabbi Meir, who stated that one may not have his hair cut by a gentile in any location.

The Gemara asks: And didn’t Rav Ḥana bar Bizna violate the ruling of the Rabbis as well? Say that when the Rabbis stated that it is permitted to have one’s hair cut by a gentile, they were referring to a haircut performed in a public domain; but with regard to a haircut performed in a private domain, did they say that it is permitted? Since the side streets of Neharde’a cannot be considered a public domain, evidently Rav Ḥana bar Bizna violated the ruling of the Rabbis. The Gemara explains: And Rav Ḥana bar Bizna maintains: With regard to the side streets of Neharde’a, since many people are present there, they are similar to a public domain, and it would therefore be permitted to have one’s hair cut there according to the opinion of the Rabbis.

The baraita stated: And in the case of a gentile who has his hair cut by a Jew, when the Jew reaches the gentile’s forelock, he removes his hand and does not cut it, because it is associated with idol worship. The Gemara asks: And how much space should the Jew leave around the forelock? Rav Malkiyya says that Rav Adda bar Ahava says: Three fingerbreadths in each and every direction.

I will leave aside the hilarious Sweeney Todd imagery of bloodthirsty barbers, counterintuitively attacking people in their home only if they're poor, for you to enjoy yourself. This section has nothing to do with Jewish barbers looking at the hairs on their patrons' heads while avoiding homoeroticism. This is about Gentile barbers slitting throats, nothing more and nothing less.
It's possible that Rabbeinu Bahya was referencing a different Talmudic passage about the law's application to barbers and their Jewish customers carefully studying each other through a mirror during a shave. I challenge you to find it for me.
אֵין רוֹאִין בַּמִּרְאֶה בַשַּׁבָּת. אִם הָֽיְתָה קְבוּעָה בַכּוֹתֶל. רִבִּי מַתִּיר. וַחֲכָמִים אוֹסְרִין. רִבִּי אָחָא בְשֵׁם רִבִּי בָּא. טַעֲמֵיהּ דְּהָדֵין דְּאָסַר. פְּעָמִים שֶׁהִיא רוֹאָה נֵימָא אַחַת לְבָנָה וְהִיא תוֹלְשָׁתָהּ וְהִיא בָאָה לִידֵי חִייוּב חַטָּאת. וְהָאִישׁ אֲפִילוּ בַחוֹל אָסוּר. שֶׁאֵינוֹ דֶרֶךְ כָּבוֹד. שְׁלֹשָׁה דְבָרִים הִתִּירוּ לְבֵית רִבִּי. שֶׁיְּהוּ רוֹאִין בַּמִּרְאֶה. וְשֶׁיְּהוּ מְסַפְּרִין קוֹמֵי. וְשֶׁיְּהוּ מְלַמְּדִין אֶת בְּנֵיהֶן יְווָנִית. שֶׁהָיוּ זְקוּקִין לַמַּלְכוּת. רִבִּי אַבָּהוּ בְשֵׁם רִבִּי יוֹחָנָן. מוּתָּר אָדָם לְלַמֵּד אֶת בִּתּוֹ יְווָנִית. מִפְּנֵי שֶׁהוּא תַכְשִׁיט לָהּ. שָׁמַע שִׁמְעוֹן בַּר בָּא וְאָמַר. בְּגִין דְּרִבִּי אַבָּהוּ בָעֵי מַלְפָּה בְנָתֵיהּ יְווָנִית הוּא תָלֵי לָהּ בְּרִבִּי יוֹחָנָן. שָׁמַע רִבִּי אַבָּהוּ וְמַר. יָבֹא עָלַי אִם לֹא שְׁמַעְתִּיהָ מִן רִבִּי יוֹחָנָן.

One may not look into a mirror on the Sabbath. If it was fixed on a wall, Rebbi permits and the Sages prohibit. Rebbi Aḥa in the name of Rebbi Abba: The reason of him who forbids that occasionally she will see a white hair and tear it out which brings her to the obligation of a purification sacrifice. But for a man it is forbidden also on weekdays because it is not honorable. Three things they permitted the House of Rebbi, that they might look into a mirror, that they got a haircut with a lock, and that they taught their children Greek, because they were in need of Roman government connections. Rebbi Abbahu in the name of Rebbi Joḥanan: A person may teach Greek to his daughter because it is an ornament for her. Simeon bar Abba heard that and said, because he wants to teach his daughters he attaches it to Rebbi Joḥanan. Rebbi Abbahu heard this and said, it should come over me if I did not hear this from Rebbi Joḥanan.

It's interesting to me that the wrongdoer's gender has changed. No longer do we have a man examining himself in the mirror for stray hairs, but a man prohibiting a woman to do so - a subtle change that (I would guess) introduces the idea of vanity by playing on misogyny.
Here we learn what Rabbeinu Bahya meant with "Seeing that they were one of three things which the rabbis had to give a special dispensation to for the household of Rabbi Yehudah Hanassi..."
However, the meaning is nowhere obvious enough for the sentence continuing, "...it is clear that generally speaking male Jews are not supposed to admire themselves in a mirror." As mentioned earlier in Shabbat 94b, we do see that the rule about plucking out hairs is gendered - women may pluck any day of the week except Shabbat, but men are not allowed.
Finally, the mirrors-make-you-girly rule! The translator notes: "But for a man it is forbidden also on weekdays because it is not honorable. This paragraph also is in Avodah zarah 2:2, Notes 143–144." Deja vu, but great, let's go!
הַמְסַפֵּר לְגוֹי מְסַפֵּר עַד שֶׁמַּגִּיעַ לַבְּלוּרִית. וְכֵיוָן שֶׁמַּגִּיעַ לַבְּלוּרִית הֲרֵי זֶה מוֹשֵׁךְ אֶת יָדָיו. הַמִּסְתַּפֵּר מִן הַגּוֹי הֲרֵי זֶה רוֹאֶה בַמַּרְאֶה. מִן הַכּוּתִי אֵינוֹ רוֹאֶה בַמַּרְאֶה. הִתִּירוּ לְבֵית רִבִּי שֶׁיְּהוּ רוֹאִין. שֶׁהָיוּ זְקוּקִין לַמַּלְכוּת. שְׁלֹשָׁה דְבָרִים הִתִּירוּ לְבֵית רִבִּי. שֶׁיְּהוּ רוֹאִין בַּמַרְאֶה וְשֶׁיְּהוּ מְסַפְּרִין קוֹמֵי וְשֶׁיְּלַמְּדוּ אֶת בְּנֵיהֶן יְווָנִית. שֶׁהָיוּ זְקוּקִין לַמַּלְכוּת.

“One who cuts a Gentile’s hair cuts it until he comes to the lock. Once he comes to the lock he desists. One who gets a haircut from a Gentile looks in the mirror, from a Samaritan he does not look into the mirror. They permitted the family of Rebbi that they might look, because they were in need of Roman government connections.” Three things they permitted the House of Rebbi, that they might look into a mirror, that they got a haircut with a lock, and that they taught their children Greek, because they were in need of Roman government connections.

Again, the translator notes do the heavy lifting for the mirror rule: "One who gets a haircut from a Gentile looks in the mirror To make sure that the Gentile will not cut his throat. In general, for a man to look in a mirror to check on his grooming is rabbinically forbidden as “woman’s garb” (Deut. 22:5). Babli 29a."
This links to Avodah Zarah 29a, meaning that we have now come full-circle.
If we read these verses repeatedly, as if making a nitpicky Sefaria sheet, we might notice the discrepancy.
We know a man cannot pluck white hairs out of his beard - possibly for gender conformity reasons or out of personal vanity - and then we are told, "One may not look into a mirror on the Sabbath [... Because] occasionally she will see a white hair and tear it out which brings her to the obligation of a purification sacrifice. But for a man it is forbidden also on weekdays because it is not honorable." We know to read this as "men cannot pluck hairs on Shabbat nor on weekdays," which is mentioned as if in parentheses, as a reminder of common knowledge.
But without that context, it seems just as valid to read that section as "One may not look into a mirror on Shabbat - women do it for vanity, and men are never allowed to do it." Then you're left trying to justify why something is okay 6/7 of the time for women but never for men, and it is somehow due to the crossdressing prohibition. You can only conclude that the act of looking in a mirror is flamboyantly feminine, which of course isn't true... but the rabbis seemed to think so.
אָסוּר לְאִישׁ לְלַקֵּט אֲפִלּוּ שְׂעָרָה אַחַת לְבָנָה מִתּוֹךְ הַשְּׁחוֹרוֹת, שֶׁזֶּהוּ נוֹי אִשָּׁה, וְאָסוּר מִשּׁוּם לֹא יִלְבַּשׁ גֶבֶר. וְכֵן אָסוּר לוֹ לִצְבּוֹעַ אֲפִלּוּ שַׂעֲרָה אַחַת לְבָנָה שֶׁתְּהֵא שְׁחוֹרָה. וְכֵן אָסוּר לְאִישׁ לְהִסְתַּכֵּל בַּמַּרְאֶה. וְאִם רוֹאֶה מִשּׁוּם רְפוּאָה, אוֹ שֶׁמְּסַפֵּר אֶת עַצְמוֹ, אוֹ כְּדֵי לְהָסִיר הַכְּתָמִים מֵעַל פָּנָיו אוֹ הַנּוֹצוֹת מֵרֹאשׁוֹ, מֻתָּר. וּבְמָקוֹם שֶהַדֶרֶךְ הוּא שֶׁגַּם הָאֲנָשִׁים רוֹאִים בַּמַּרְאֶה, בְּכָל עִנְיָן מֻתָּר.
A man is forbidden to pick even one white hair from among the black ones, for this is the way women beautify themselves, and is forbidden because of the enjoinder that "A man shall not wear (a woman's garment)." It is also forbidden to dye even one white hair to make it black. A man is also forbidden to look into a mirror. However, if you look into it for medical purposes, or when you are cutting your own hair, or wish to remove stains from your face, or remove feathers from your head, it is permitted (to use a mirror). In a region where it is customary for men, to look into mirrors, it is permissible for any reason.
In searching for the last bit of contentious text - where Rabbeinu Bahya heard that it is permitted to use a mirror for medical purposes - I have only been able to find this.
The issue is that the English version is a copy of a copy of a copy, so I can't check the original. This quote from the Kitzur Shulchan Arukh (“Abridged Shulchan Arukh”) is, per Sefaria, "a simplified summary of the Shulchan Arukh." It was written in the mid-1800s CE.
So let's go to the Shulchan Arukh, which was "compiled in the 16th century by Rabbi Yosef Karo, it is a condensed and simplified version of the Beit Yosef"... darn.
At least the Beit Yosef was by the same author in the same time period - published no later than 1559, and - it's a commentary too, based on Arba’ah Turim, "a 14th-century legal code by Rabbi Jacob ben Asher" published in the mid-1300s CE. Is this the primary source? Who knows! I'm dizzy!

Since I can't read it, I can't check what he referenced directly, or if he made the logical leap himself (since looking in a mirror must be permitted if it would save a life, and might therefore be allowed in all questions of health).
I will include the Hebrew below in hopes of coming back to this someday.
הזורקים חטים לפני חתנים צריך ליזהר שלא יזרקו אלא במקום נקי וגם יכבדו אותם משם כדי שלא ידרסו עליהם:
ואהא דתניא זורקים לפניהם קליות ואגוזים בימות החמה אבל לא בימות הגשמים כתבו התוספות ולא בימות הגשמים אע"פ שבאגוזים אין האוכל נמאס בתוכו מ"מ כשהן נופלין בטיט נמאסין ומשמע מדבריהם שאפילו כשהם בקליפתם אין זורקין אותם בימות הגשמים: וכתבו עוד התוספות דעכשיו שדרכן לזרוק חטים בבית חתנים צריך ליזהר שלא יזרקום אלא במקום נקי וכ"כ ה"ר יונה וכ"כ בהגהות מיימוניות פ"ז וכתוב עוד שם שמהר"ם צוה לכבדם משם כדי שלא ידרסו עליהם ולא נהגו כן: כתב הרוקח בשם מ"ס ראב"י אומר כל אוכל שגמרו בידי אדם מותר לכסות בו ראב"י אומר כל שיש לו יד ועוקץ מותר לכסות בו ע"כ ומשמע דאפילו מאן דפליג על שמואל מודה דבכה"ג שרי להשתמש באוכלין:
אמר שמואל עושה אדם כל צרכיו בפת והלכתא כוותיה וה"מ במידי דלא ממאיס ביה אבל במידי דממאיס ביה לא הילכך אין מניחין עליו בשר חי ואין מעבירין עליו כוס מלא ואין סומכין בו הקערה אי ממאיס בהכי אבל אי לא ממאיס ביה שרי: ואין נוטלין הידים ביין בין חי בין מזוג אפילו נטילה שאין צורך אכילה ואין זורקין הפת משום בזיון אוכלין וכשם שאין זורקין את הפת כך אין זורקין כל האוכלין הנמאסים על ידי זריקה כמו תותים ותאנים אבל במידי דלא ממאיס כגון רימונים ואגוזים שרי: כתב א"א הרא"ש ז"ל יש נוהגין לאכול דייסא בפת במקום כף כיון שאוכלין הפת אח"כ ובמסכת סופרים גרסינן אין נוהגין ביזוי באוכלין אין זורקין אוכלין ממקום למקום ולא ישב אדם על קופה מלאה תאנים וגרוגרות אבל יושב הוא על קופה מלאה קטנית או על עיגול של דבילה מפני שנהגו כן אין סומכין באוכלין ואין מכסין באוכלין ואין אוכלין אוכלין באוכלין אא"כ היה ראוין לאכילה פי' אלו עם אלו ואוכלן ביחד ולפ"ז אסור לאכול דייסא בפת במקום כף ואפשר דההיא דמסכת סופרים פליגא אדשמואל דקתני אין סומכין ואין מכסין באוכלין ולשמואל שרי דהא לא ממאיס ואנן קי"ל כשמואל ולא ממאיס הפת שאוכל בה דייסא ואני ראיתי גדולים שהיו אוכלים בכל פעם מעט מן הפת עם הדייסא עד כאן: ממשיכין יין בצינורות לפני חתן וכלה והוא שיקבלנו בכלי בפי הצינור וזורקין לפניהם קליות ואגוזים בימות החמה שאינו נמאס אבל לא בימות הגשמים מפני שנמאסים ולא חתיכות וגלוסקא לעולם: