I think that she tends to right with her right hand, though. Pics of her writing with her left hand are usually the mirror images (for example, the Love~Destiny PV).

I'm gonna take a guess and say that she is ambidextrous because she started out left-handed, but in Asia being left-handed is discouraged. Parents then try to get their kids to write with the right hand, and so one becomes ambidextrous (I figure this is the reason my cousin is ambidextrous, maybe Ayumi also went through this)