You can craft or buy 'paint' that will overlap/remove faction insignias, so you can wear them wherever you please. Example, the insignia on Brotherhood Power Armor can be chipped off. And I think that Enclave Power Armor should be hostile to NCR/Brotherhood until paint is applied.
I've always wanted to wear Centurion armor with T-51b Helmet (looks incredible!) but I can't, due to Legion haters.