Why Zuko sided with Azula over Iroh in Ba Sing Se (and why he was somewhat "right" to do so)
One of the "twists" in Avatar that make more and more sense the more I think about it, was Zuko betraying his uncle when Azula gives him a chance to come home - and it is also one of my favorite parallels in the narrative.
In "Avatar State" we see Azula suddenly coming back into Zuko and Iroh's life, telling her brother the very lie he wanted so desperately to convince himself was true (that Ozai was genuinely sorry and wanted him around), while Iroh was trying to snap him out of it.
However, as the season went on, we see that dynamic start to shift, and Iroh is the one who is clearly struggling to see the obvious: that Zuko would never be truly happy leading that simple, honest, good life they had in Ba Sing Se (at least not without getting some kind of closure/truly understanding just how badly his father hurt him).
Iroh wanted the best for Zuko, and he genuinely loved him, but he was seeing only the good in him, which made Zuko vulnerable to his most toxic traits (impatience, entitlement, selfishness, exaggerated pride and anger). He was ignoring the simple facts that:
1 - Even though Iroh obviously like a father to him, and Zuko loved him dearly, he still loved Ozai and wanted his love as well.
2 - No matter how great of a life he was offered in Ba Sing Se, Zuko did not chose it, but was forced into it in a very traumatic way, losing his family, his home, his country, and literally everything he knew, and being thrown in enemy territory, having to either keep chasing an "impossible" goal of capturing someone who was "dead", or to just accept that the life he knew before was over and that he would never even see his father again, let alone be able to reconciliate with him OR call him out for his abuse.
Iroh was keeping Zuko safe and trying to help him make the best out of a bad situation. He was also ignoring his emotional needs. And when Zuko gave in and started acting like the son he thought Iroh wanted (that was so completely un-zuko-like that it was almost disturbing to watch), Iroh didn't recognize how false and unhealthy that was.
Then Azula comes back, and just seeing her, just having one small bit of his old life back immediately makes the real Zuko come back.
Unlike their uncle, Azula did not choose to see only one side of her brother. She knew very well that, despite of everything he had gone through and how much he had changed, he was still the proud prince of the Fire Nation (something Zuko himself says out loud in "Zuko Alone" right after remembering his mother telling him "Never forget who you are") and that he was still the loyal son that wanted nothing but to be someone his father would love and be proud of.
There's also one thing that people forget that Azula was implicitly offering to her brother on that moment: her approval. Their relationship was a train-wreck, but Zuko had always (reluctantly) admired Azula, and on that moment she was honest with him. She promised she'd take him home if he helped her, and she did. She said he restored his own honor that night, and she meant it. On "The Headband" she told him she was looking out for him despite gaining nothing by doing so and she was telling the truth. On "The Beach" she knew he was upset, knew where to find him, and tried to cheer him up (in a unhealthy way, but this ain't the point). Zuko's relationship with sister is much more complicated than Iroh realized, and they did care about each other, in their own weird, dysfunctional way - which was another factor on Zuko's betrayal, after all, he only joined the fight when Azula was being cornered by Aang and Katara.
But most important of all, she did something Iroh thought he was doing for him, but wasn't able to truly put into practice: allowing Zuko to choose. "Who are you? And what do you want?" Really important questions that Zuko needed to figure out the answers to, and that Iroh convinced himself he already knew. Zuko wanted to do right, and he didn't want any innocents to suffer, but he never stopped wanting the throne and the approval of his family. Only by truly seeing the appalling things his father, his sister, and his entire nation were willing to do to the rest of the world (and with Iroh no longer coddling him, but instead holding him accountable) did he finally understand that he could either be a good person, or he could keep supporting them out of blind loyalty. He would never be able to do both.
Azula allowed her brother to do one of the worst things he could have ever done and to reach his lowest point - and that allowed him to choose right, not be forced into acting a certain way. Ironically, she, the perfect Fire Nation soldier, Ozai's "evil" daughter and favorite weapon, was the one who saved Zuko - or rather, showed him the way to save himself.
(Too bad those goddamn comics chose to have Zuko abuse her while she was at her most vulnerable instead of trying to do the same for her)