well whatever it is, my body knows exactly how to use a two (or one and a half) handed weapon without any previous training or knowledge other than the basic slash, stab, and chopping actions.
No, it's not that weird. It just comes more naturally to some people - others would have a harder time with it, but would have less trouble with one-handed weapons.
I believe it has more to do with less tangible factors like psychic, a preferred way to move, or even your clothing - many of them being often underestimated or not considered at all.
Wakizashi is typically worn beside/over the katana. At least in every drawing/painting I've ever seen from classical Japan. Both go edge up ("blade up" is physically impossible, if the sword is to stay in the scabbard), because it is easier to draw the sword in a fluid manner thanks to the curve--to draw and cut in one motion--as the scabbard is basically tucked into the belt, rather than hanging from the belt in the Western style.
Yeah, I was looking for "edge up", but didn't remember the idiom. Sorry for that, sometimes my limited mastery of English leads to such blunders.
For the record, I have been shown the turning of the scabbard that Oniya described - I don't know how "historical" it is, but it looks a good way to draw the blade fast.