I very much doubt if it's cultural. Of course attitudes to it are -I imagine there are cultures where left-handed is regarded as threatening (which have usages like "sinister") and probably others where it's seen as some kind of advantage. Which it can be in fighting and probably for some other things. But my understanding is that it's pretty clear that whether you are left-handed or right-handed is genetically determined.

One thing I've always been unclear about is how this all ties up with the brain hemispheres - I can't remmebr how it goes, but one hemisphere is specialised on some things and the other on other things.

Well, if you're left-handed is that reversed - or does it stay the same , but you are stronger on the stuff that the appropriate brain hemisphere specialises in. Or does this vary, with some left=handers being wired up to a left-handed brain, and some not?