I can totally see Zuko and Azula having some moments when they get all snappy at each other (be it just because they're petty or for some good reason - usually both) and there's this weird tension, but also a bit of tenderness.
Sort of like that scene in Disney's Beauty And The Beast, after Beast saves Belle and she's taking care of his wounds and he is being a dick to her. Azula has 100% told Zuko a billion times to control his temper and Zuko has definitively had some moments where he KNOWS Azula has a point, but he just refuses to admit he did something wrong - he especially doesn't want to admit to her - and he has to constantly keep trying to find some excuse to shift the blame back at her (he has also 100% snarled, growled and howled at her when angry, don't at me)
And obviously, these two had many, many, many, many, MANY moments when, right after a fight, one of them just… lets their guard down for a second and the other just can't help but do it too. And considering their lives, it is very likely that this happens when one saves the other, because they just cannot pretend they don't care.