Hello nader,
I have had several horses that came to me with a tendency to be reluctant picking up a particular foot but did not have current pain issues. I have always thought it likely the horse had some problem with that leg/foot in the past and became adverse to having it messed with. Even the reverse is possible where the opposite foot was painful therefore the horse became reluctant picking up the good foot as it hurt.
Past lameness problems or congenital digital flexor issues cause changes in foot shape and hoof angle to be asymmetrical. If this is the case the hoof with the more narrow heel and steeper sloping hoof has a tighter digital flexor causing a relatively lighter loading of the heel on that side. Perhaps this asymmetry is the cause of the changes in way of going.
Assuming there are no lameness issues (have a veterinarian examine him) the sidedness you describe is a frequent issue with young horses in training and once lameness is eliminated designing workouts that deal with such issues will overtime help control these tendencies.
DrO