My understanding stems from the human anatomy, but it most likely corresponds with deer as the central nervous system of mammals is very similar across the board.
In the brain stem of mammals, the external stimuli that occurs below the brain stem (meaning that the source of stimulus affects the nerves that connect to the spinal cord or directly to the brain, i.e. touch) then the stimuli is actually crossed to the opposite side of the brain for processing. This is why left-handed people are referred to as "right-brained" and vice-versa. So this principle also is reflected in injuries. If an animal is injured on one side below the base of the neck (most soft-tissue injuries) then the opposite side of the head (and therefore antlers) will be affected. This is why those whom have suffered a stroke in the right side of their brain, they will have their left side extremities affected, while the right side of their face is affected and vice-versa. However, if an injury, weather visible or not, occurs above this part of the brain stem like taking an antler to the face during the rut or a predator attack, then the same side of the head (and therefore antlers will be affected. If the injury affects bilaterally (meaning both sides of the body) then it will be reflected to both sides of the antlers.
The tricky part is that when an animal is injured, they may favor the opposite, stronger side of the body which can lead to injury or more rapid degeneration of the favored side and in future years both antlers can be affected.
Another thing to remember is that some injuries are not manifested visibly. There can be damage that is not apparent to the naked eye on one side of the body, while the opposite side may have a very apparent injury present.
I hope this explanation helps, but it has held true in the deformed antlered animals I have seen personally.