I've been mounting my horse from the 'wrong' side for years due to a knee injury. At first I'd try to mount from the left, not even thinking until I felt pain. I still dismounted to the left, and mounted bareback from the left.

The knee is more or less okay now, but I still mount from the right and dismount to the left. Any time someone new sees me hop up on a horse I get the strangest look!

It may be easier for me to switch sides as us lefties are forced to use our less-dominant sides in this right-handed world.

On a bicycle I make a point of it to build balance by pushing off with the 'wrong' foot, coasting with the wrong foot forward, going one-hand with my less balanced side.

When running, I leap up onto cement barriers with the wrong foot and land on the wrong foot as much as possible (it was the dominant landing leg that got me the aforementioned knee injury in the first place).

Don't go past any safety boundaries, especially with clipless pedals, but do challenge your comfort zones whenever possible. With a little practice you can train your less dominant side.