Grasscarp has the best answer... but from the American perspective (as well as horse country here in the western U.S.) it doesn't have to go back to medieval days. The U.S. Cavalry, (and most other countries) since it's inception, wore their sabres on the left side (again, primarily because most are right handed) and mounting the horse from the left keeps the sabre and its scabbard out of the way.
Still can't picture anyone getting on their horse in the manner hc4361 prescribes without winding up facing the ass end of the horse...
Additionally, cataract surgery is exremely common here in the U.S. In and out in 30 minutes, vision quality greatly improved (near 20/20)...surely the NHS does this in the U.K., no?