Pins and keys should be removed from the same side as inserted or they will wedge and possibly crack the wood. Most are tapered.
One responder on my muzzle loading forum did remind me that on C&B revolvers (Colt style) the retaining wedges do go in from the left.
BTW: left or right refers to the shooters L or R when aiming the rifle.
I don't own a good Hawken repro (or original for that matter
) but the builder of my Jaeger has one and is one of, if not the best, maker of repro Hawkens there is. I'll see him later this summer and will check it out then. I might even phone him, been a long time.