I'm left hip biased - according to Amira al Kattan "This is the Egyptian way".
Something tells me that if I was right hip biased, THAT would have been the Egyptian way.
I think everybody has a strong hip and a weak hip - I think there's always a disparity (which seems unrelated to your writing hand bias or left/right brain bias or anything else you care to think of!). However, I think it gets worse if we have an injury that causes us to then have to favour one side for a while - or just that laziness thing where you use the good hip to look impressive!
For me, the only thing that help is PURPOSEFULLY practising on the weaker hip. Nothing particular, just the usual moves I would normally do - but starting on the right, finishing on the right, repeating on the right, travelling to the right, etc etc. I still feel my left is much stronger than my right, and when I improv (which is most of the time), everything comes more naturally to the left and as such I use it more. I need to think a little about the right, keep it in mind that it's there. I feel like such a CLUTZ when we do right-sided things in class, because my balance is worse and my transiotions are rougher.
So if I'm left to myself, in class I also favour the strong hip... and allow myself to practise clutzily on the right at home.