There is something that happened on 9 Adar that is more than simple disagreement -- it is a time when the houses of Shammai and Hillel separated. There are times when we see Hillel and Shammai as incompatible truths, and other times when they are seen as two legitimate methods of interpreting Torah. However, their disagreement on 9 Adar led to the declaration of a fast day (not regularly observed today). Because of this history, the Pardes center for Judaism and Conflict Resolution began an annual day of learning on 9 Adar.
Check out this website https://www.9adar.org/. This is the website from this initiative to foster "constructive conflict" within the Jewish community. What does it mean to have "Constructive Conflict?" What does it mean to have "Destructive Conflict?" Here's an excellent source sheet with a LOT of resources: https://elmad.pardes.org/wp-content/uploads/2014/01/The-Story-of-9Adar-A-Day-of-Disagreement.pdf
This is the classic example of Shammai and Hillel respectfully disagreeing. The Talmud even goes so far as to explain that the reason why the halakha (Jewish Law) is according to Hillel is because of his approach toward disagreement. We see this in even greater focus in the following text from Mishnah Yevamot
This mishnah draws an even sharper focus on the disagreements between Hillel and Shammai. Not only did they disagree with one another over significant issues, they made conscious decisions to remain in community and relationship with one another.
This text does not paint as pretty a picture of Hillel and Shammai. This shows them as just the opposite. This is a day where such deep disagreement -- and the method of disagreement -- literally brought violence into the house of Torah study. This is why the later sages declared 9 Adar as a fast day because it led to a total separation between Shammai and Hillel.
זכריה פרנקל, דרכי המשנה, עמ' 41- 44 והנה עד ימי הלל ושמאי לא נתפשטה מחלוקת בהלכה, וגם הלל ושמאי לא נחלקו כי אם בג' או בד' דברים.... ואחרי מות שמאי נתרבו המחלוקות ונתלהבה אש המריבה והעלתה אף וחרון בלב תלמידי ב"ש. וזה לרוב דרך תולדות דעות חדשות, בימי התהוותן יעלו קנאה וחימה בלב מולידיהן נגד בעלי דעה אחרת. וב"ש הפריזו על מדת רבם אשר עמד בתוקף נגד הלל וביקשו לדחות ב"ה )וגם את הלל עצמו(מלעשות על פי הלכתם, וגם קמו בחרבות וברמחים להשבית דעת החולקים ולהקים ההלכה הישרה בעיניהם... וברבות השנים שבו ב"ש מדעתם הקשה ... והשיבו על לבם כי לא בכח יגבר איש העוסק בתורה לשמה כי אם במשא ומתן של בינה, והרשות נתונה לכל חוקר לדרשות ולחוות דעתו. אז נתחזק השלום בינם ובין ב"ה, והיו נוהגין זה בזה אחוה ורעות.
Zecharias Frankel Darcheh HaMishnah, 1859, pp. 55-56. And behold, until the days of Hillel and Shammai disagreements of the law did not spread, and even Hillel and Shammai did not disagree except for on three or four matters…. And after the death of Shammai, disagreements increased and the flame of conflict was ignited, and anger and rage increased in the hearts of Beit Shammai. And this normally occurs when there is the inception of new opinions. And Beit Shammai went even further than Shammia himself who stood firmly against Hillel and they attempted to reject Beit Hillel (and even Hillel himself) from practicing (Jewish law) in accordance to their rulings. They also rose up upon them with swords and spears to eliminate the opposing opinion and to establish the law as seemed right in their eyes…. But over the course of time Beit Shammai returned from their rigid way, and returned to their hearts that not through force a person should be victorious when dealing with Torah for its own sake, rather through thoughtful negotiation. And permission is granted to anyone to research, teach and express his opinion. Then the peace between Beit Shammai and Beit Hillel was re-strengthened, and they treated themselves in brotherliness and friendship.
This text is written by Zecharias Frankel, the founder of what would later become Conservative Judaism, in Germany. It is from his commentary on the Mishnah and is translated by Daniel Roth. See how Frankel tries to find synthesis between the Hillel and Shammai who bring violence into the Beit Midrash and the Hillel and Shammai who live together in the community of Israel.