I've heard that the stairs in castles all spiral the same direction so the disadvantage from being higher up on the steps is counterbalanced by having more space to swing your sword arm without blunting it on walls or the steps. Or at least if you're right-handed.
They taught us this in fencing as a side discussion (like during a break). Apparently it isn't true, as other commenters pointed out.
The button side of jackets thing is true though, except usually also told wrong. We're often taught in fencing that men's jackets button left over right so that a sword drawn by a right handed person doesn't fuck up your own clothing. That's also wrong, but close: men's jackets button left over right because you traditionally put your shielded side forward, your left side, and you wanted any armor pieces to overlap so that there's no holes to get caught in if you're poked from the left. Which is a stance backwards from later style one handed fencing. The not messing up clothes on the draw of a sword is just extra bonus from the original reason.
If I recall correctly, that theory is challenged by: (A) there isn't a predominant direction to such staircases and (B) if enemy soldiers are in your staircases, it's all too late anyway.