Samurai Mount a Horse From the Right Side

While [Western] equestrians mount their horses from the left side, the samurai always mounted a horse from the right side. One of the reasons being that the sword, in this case the tachi, was worn slung on the left, and while wearing armour, the shorter wakazashi companion sword was tied firmly into the obi sash to the left. Mounting from the left risks catching or hitting the tsukagashira sword handle end caps on the saddle. Mounting from the right keeps one’s hips and shoulders pointing in the same direction as the tsukagashira on one’s wakazashi or daisho (paired katana and wakazashi).

— The Samurai Castle Master, Chapter 2