Thanks for posting that pic Trevor! Now everyone is going to tease me about my 1968 Miley Horse trailer.
I would not put the gun butt backward. Too many guys get back to the truck and the gun is gone- pulled out on the brush.
I suggest butt forward, scope up, on horses right side.
Four reasons:
1) Don't let a gun ride scope down. Gun is a 7 pound anvil resting on your optic as Mr Wiggles jars it with every step.
2) If you're a lefty, while mounted, you can reach across the horse's neck, grab the pistol grip portion of the stock, squeeze your right leg against the scabbard, and pull the rifle out as you step off the horse, prefferably to his right. It's quick and you'll get a shot off while your buddy runs around the back of his horse to pull his gun out. Also, he can go catch the horses after they run off because your busy gutting your buck.
3) The bolt will face away from the horse.
4) You're a lefty, so the scabbard won't be in the way of the cinch. I'm a righty, so mine is, but I just fight it for the reasons posted above.