While no one will ever convince me that Azula is not Izumi's mother, I do like to imagine what would it be like if Izumi's mom really was some random woman Zuko had a fling with and either died or walked off, and Azula essentially became her step-mom before she ever got together with Zuko.
Like, Azula would have already gotten plenty of therapy, her relationship with Zuko is much, much better (though they still argue and tease each other because of course), and obviously she'd want to go home, and Zuko is struggling to handle both his responsibilities as Fire Lord AND as a father, and he trusts Azula now and she's great with Izumi, so he brings her back to the palace and gives her political power.
Everyone around would be uncomfortable as all hell because Zuko and Azula just cannot help but act incestuous around each other (despite having never done anything with each other and maybe not even being aware of their feelings) and Azula just loves Izumi so much, and since that goes against her image of "cruel, crazy princess" people start wondering if she really is just Izumi's "evil" aunt. Izumi's happiness when they finally get together only causes people to speculate even more - especially because Azula is constantly hinting that maybe she is her mother once she hears about that rumor.