When I hunted on horseback (always dismounted to shoot!!!), I packed my scope-sighted, bolt-action rifle, butt rear, sights up on the right side. I made my own scabbards, and had a flap that covered the rear end of the scope, plus a retainer that fit over the bolt. The scabbard was angled so that the buttstock stayed below the level of the cantle, so as not to interfere with my right leg as I mounted or dismounted. The problem with having the stock butt forward on either side of the horse was catching brush when riding through timber. In addition, with the "southeast" carry, I could draw the rifle before dismounting or while on the ground.
I remember seeing one of The Magnificent Seven mounting and catching his leg between the rifle scabbard and the horse. He was able to disengage and finish mounting. However, part of these problems had to do with the buttstock sticking up above the level of the top of the horse. Never had a problem with the way I attached mine.