Most people don’t put any thought into why they’re taught to post on the outside diagonal — when the inside hind leg and outside shoulder are reaching forward in the trot — other than it seems more comfortable. Certainly it’s heresy in some disciplines to be caught on the “wrong” diagonal.
This emphasis on a “right” and “wrong” diagonal is mostly a factor of riding in an enclosed space and thus constantly needing a curved arc for every corner. When trotting on a curve, one hind leg covers more distance and thrusts harder than the other, and thus rising for that increased thrust is easier on the rider and also on the horse if the curved line is maintained for any length of time.
The outside diagonal is therefore usually the “right” diagonal. Gradually increasing use of the inside diagonal, however, can be an effective tool to build more strength behind, especially when one hind leg is weaker than the other.
The location where you change diagonals should also be determined by where it most benefits the horse. When riders change diagonals across the ring, they usually do so somewhere near the centerline. However, it will aid the horse’s balance to change the diagonal when the bend is changed. If you change the bend coming out of the first corner, going into the second corner, or somewhere in between, then that’s where you should change the diagonal.
The concept of a “correct” diagonal is disregarded when showing dressage, and there’s no reference to it in the dressage section of the USAEquestrian Rulebook. Dressage riders shouldn’t be penalized for posting on the “wrong diagonal” unless the judge sees that the trot is unlevel. On the other hand, posting on the inside diagonal can loosen the back of a horse that is particularly stiff on one side and thus help his score.
Posting was invented as a way to relieve the effort of the horse and rider covering long distances at the trot. Riders going “post to post” would change diagonals every few minutes so one hind leg wasn’t working harder than the other. This concept is why posting on the inside diagonal might be a benefit.