Traditionally, we do stuff on the left side of the horse. Lead, bridle, saddle, mount, dismount, etc. Tack is made to be handled from the left side. Blankets, halters, bridles, girths and more are all designed to be fastened on the horse's left side. And then we ride and we say to our horses: Disregard everything I have done to this point, now you must be equal on both sides, both leads, both directions.
The left side thing is something people invented. Horses don't care. They don't care, unless they've only had things done on their left sides, that is. When I work with horses, I make an effort to do everything on both sides as much as possible because I don't think it's fair or healthy to only work from the horse's left and then expect him to be ambidextrous in the ring.
Leading horses from the left was developed because most people are right handed so then they would be holding the horses with their dominant hand. Left handed people were out of luck. Mounting & dismounting on the left was started by the knights who were right handed and wore their sword scabbards on the left. It was pretty tricky to mount a horse on the right by swinging a left leg encumbered by a sword up and over the horse without leaving a mark. Plus, since they were already leading their horses on the left, it didn't make much sense to walk all the way around to the other side to get on.
Which is how I came upon the idea of doing things on both sides. Sheer laziness. One day, tacking my horse up in the barn aisle-way, I picked up my saddle from the saddle rack on the right side and considered how I was supposed to walk all the way around to the other side to put the darn thing up. That's just silly. Then it just made me think about the silliness and impracticality of working on the left in general.
I suppose the idea actually started to from way back when I was in college. A friend of mine was graciously allowing me to exercise her Appendix mare. There was one caveat - she would take off as soon as you get a foot in the stirrup. Not having enough experience or knowledge to know how to deal with that, I would cheat by getting on from the right. She wasn't expecting that and so had no anticipation or flight instinct. It worked for me and for her. Not only was it safer for me and anyone else riding in the arena at the same time, but it was certainly healthier for her back. Always mounting from one side means a consistent pull in one direction on that horse's back muscles. It also leads to one stirrup leather getting stretched out more than the other and even twisting in the saddle tree. Not only that, but from the point of view of rider fitness, you'd have stronger quads on the left from the thrust of lifting yourself into the saddle and one hip that was more flexible than the other from lifting over the horse's back.
Can we, as compassionate riders, ask our horses to bend equally, stretch equally, carry weight evenly, while we ride with one strong leg, one loose leg and one dominant hand? That smacks of irresponsibility to me. Not that I'm perfect (far from it) but I try. I do find myself cheating when something is difficult and switching to my right hand, but I try to recognize that and even out.
So for your horse's sake, and your own as well, be responsible and possibley lazy, and saddle up from whatever side of the horse to which you're closest, lead from the left and the right, and try to see both sides of every issue.