balsa-margarita asked: Does Azula like it when Zuko touches her hair, and vice versa? Will either of them even let anyone else touch theirs?
They both love having their sibling touch their hair, wash it, comb it... they even learn to cut it the way the other likes so they can do it for them. It is definitively part of their love language and is another form of intimacy and affection for them. Add in how important hair is to them (and to Avatar in general) you get an idea of just how much of a big deal this is to them.
If the other, for whatever reason, isn't there to help them with their hair, they might let the servants do it, but it's kinda like undressing so the doctor can examine you properly, it doesn't really "count" and can even be a little awkward for them.