Zuko and Azula have the most interesting relationship in the series. They grew up together, know each other unbelievably well, and were once happy. They were also raised to be enemies and nearly killed each other.
There's a reason their Agni Kai is seen as the best fight in the show. The emotional connection between them is so strong, it is almost palpable. The pain, the loss, the tragedy, the inevitability of it all… and the love.
They didn't want to care about each other anymore, and no one around them wanted them to.
Your brother is a failure.
Trust is for fools.
She's crazy, and she needs to go down.
And yet, they couldn't let go of each other.
Azula was the one the Dai Li were loyal to, but she did bring Zuko home like she said she would. She backpaddled and threatened him with the information that the Avatar was still alive… only to then tell him to be careful when seeing Iroh, instead of using that against him.
Zuko didn't care that Aang would have to kill Ozai, but was obviously conflicted when seeing Azula falling "to her death". She saved herself - of course she did - and he went back to resenting her… only to then watch with pity as she was crying after being defeated.
They were raised to be enemies, but made quite the team. They conquered Ba Sing Se together, and he helped her win that stupidly intense game at the beach. They avoided each other, but knew just how the other would act, in battle and otherwise.
They hate each other… yet there seems to be something hiding just below the surface.
Azula represents the embodiment of temptation in his mind - the blue dragon wrapping it's body around him, whispering in his ear, controlling him, corrupting him.
And she tempted him in the real world too. On the fateful, infamous night he went to her room. They were all alone, and she was so close - too close. Her voice, her movements, her stare… everything was as far from a normal sibling interaction as possible. It is all there.
But most people will never see it for what it is. And for what it could be.