Many of the practices we accept without question as "correct" when dealing with horses have been passed down to us for generations as "traditions". Perhaps the most well known tradition is that of the designation of 'far' and 'near' sides to a horse.
Once upon a time, somewhere in Europe, somewhere deep in the past, there was a king who determined that he could mount his knights faster than the neighboring kingdoms if he made a few adjustments to his knight's horsemanship and equipment. First of all, he decreed he only would have right-handed knights. That way they would all carry their swords on the same side - the left hip. This worked very well for marching, as his knights did not have their swords banging against one another should they march next to a left handed knight. Of course, that also meant that all of his knights would mount their horses on the left side so as not to have to swing their sword up over their horse when they mounted. To make the whole process even smoother, the king decreed that all of the buckles and fasteners to adjust and secure the saddle and bridle would also be on the left side. In that way, the knight could pretty much do all of his saddling and bridling from the same side and be on his horse that much quicker. Obviously then, the side of the horse where the knight stood while saddling was the "near" side, and that other side was the "far" side.
Traditions die hard. Even though I haven't ever run into a rider wearing a sword, I was taught from the start that you always mounted on the near side and to do otherwise just showed your ignorance. Not wanting to appear ignorant, I have correctly mounted from the left side for 35 years. And I lead from the left, as any good horseman knows to do. But does the horse really care? Only because we condition him to care!
It makes no difference to a foal whether you walk up to his left or right side, it is all the same to him. However, with our conditioning to be "proper" horsemen, we tend to approach all horses from the left, and soon, the foal grows to expect it. As horses are creatures of habit, the habit soon forms that the most activity is on the left. That is where the human is, the halter is buckled, the grooming starts. That is the side the saddle comes from. The side the rider crawls up on. To do it any other way eventually becomes uncomfortable to the horse, and confirms to the human that yes, there is a "right" and "wrong" side.
The only thing "wrong" about this scenario is that we allow it to happen. There is no good reason, beyond old traditions and right handed humans, that horses should be handled exclusively from the left. We create so many problems both with horses and their riders by adhering to this convention. The horses become left-eyed. They grow to prefer handling on the left and refuse to be led from the off or far side. They will not allow mounting from the wrong side. They are hard to work on the lounge line with the handler on the right side. For the rider, leading a horse left handed can be uncomfortable and mounting with the right foot in the stirrup can be impossible. Yet, if you spend any time around horses you will find the need to do this will most likely arise at some point. The most common example that comes to mind is the trail rider who needs to dismount for whatever reason on a side hill trail. Not only is it easier to mount and dismount on the up hill side, it is also safer. If the up hill side is on the right, you either have to do it on the "wrong" side or turn the horse around - which may not be practical. No matter what the occasion, should you find yourself in a situation that demands working, leading or mounting on the off or far side of the horse, you will be better prepared for it if both you and your horse are accustomed to it.
Forget tradition, and pay no attention to the people who may point and say, "Hey, that's the wrong side"! From the earliest days of your horse's life and your riding experience, develop an ambidextrous approach to horse handling. Accustom your horse to being led, saddled, mounted and dismounted from either side equally. If you already are stronger in your left leg from years of "doing it right", set a goal to mount only on the right side until your right leg is equal to your left in hefting you up into the saddle. You might even discover that the equal strength in your legs from mounting on both sides will help to make you a more balanced rider leg cue-wise as well.
One word of caution: In the show ring, traditions still reign. Lead your showmanship horse from the left and mount and dismount your equitation horse on the near side as well. Just know that you are doing it for the sake of tradition, not because your horse really, or should, care.