I must be a weirdo, because when schooling if on the right rein, reins go to the left, if on the left rein, I flip them over the the right. If carrying a whip I always change it into my inside hand and I don't like the whip and reins on the same side incase I get in a kurfuffle. On hacks they go on the left, but when riding the racehorses with a single bridge, they go on the right
But, but... the sword is on your left leg when carried, so surely when drawing and replacing your sword, reins to the left would get tangled? Although I guess that's preferable to stabing your reins rather than your opponent when carrying the sword
I am right handed, ambidextrous with whip (thought everyone would have to be if you ever school or go past scary objects?) and carry my reins inside my right hand. I noticed it a couple of months ago and tried to get used to it on the left just incase I ever needed to, and it feels wrong - I can't use my left hand like that!
I honestly haven't got a clue - and have never thought about it either. But sitting here trying to visualise, I've got a feeling my rein loop might fall to the left (I'm right handed) and definitely under.
Also, re. drawing a sword (I have actually ridden with swords many times in the past but that's another story!). The sword would be carried on your left side and drawn with the right hand. The reins would be held in one hand only - the left hand. You would move your right hand OVER the left, rein holding hand, in order to draw the sword - so a rein loop falling underneath, on the left side, is actually prevented from being caught in the sword by the left hand (iyswim).
but saying that i did notice that i tend to have my right hand nearer the center of the horse - so when i school i put them on the right (under) and i find the difference in feel makes me put my right hand in a better position....