The horse may begin the trot by moving his right hind and left fore together, then the left hind and right fore together. There is no overstriding , and the horse must shorten his body and elevate the flight of the hoof. The jog is essentially a very short forward movement while the trot should have good extension. The left hind and the right fore then move together. The left hind is an extended stride while the right fore is a short stride.
Horses also pace, which means they move both legs on the same side at the same time.
Some horses single-foot. Gaited horses essentially walk, trot and canter, but with very exaggerated and flashy foot flight. There are some naturally gaited horses, although the excessive action seen in show competition is man-enhanced. The flashy action of most competition gaited horses is the result of painful training and shoeing techniques, not nature. So the horse knows where his feet are supposed to go.