If you watch a dressage or even hunter horse coming toward you, you might see his muzzle moving left and right in a rhythmical fashion. A few different schooling techniques, some good, some bad, can cause this type of motion.
The good schooling methods include give and take on the inside rein and bend, counter bend exercises.
Give and take on the inside rein is ridden while maintaining consistent contact on the outside rein and giving with the hand forward for a few steps on the inside rein. Performed at the walk, trot and canter, the exercise does a few things: it prevents the horse from laying in the rider's hands and balancing on the rider, it helps the horse learn and perform self carriage, and it keeps the horse from falling on the forehand. It also helps a horse that has grown dull in the mouth because of a rider's heavy hands. Additionally, the exercise helps a rider have a more sensitive feel of a horse's mouth, especially a green horse, and it helps riders develop better their independent use of aids.
The other reason a horse's muzzle might move left and right is when the rider is asking the horse to flex inside and then counter flex outside. Unlike the very small movement in the give and take mentioned above, in the bend and counter bend exercise, the horse's head and neck move slowly to the left and right while spending a moment in the center to rebalance before flexing the other way. Bend and counter bend exercises help supple horses and helps to prepare them for lateral work.
When the left and right muzzle movement is incorrect is when the rider uses a see-saw technique to force the horse into flexion at the poll. Instead of riding the horse back to front over his back as he should, this rider is riding the horse front to back and creating false flexion (not a true flexion). Instead, the horse in this scenario is evading the discomfort created by a hard rider's hands.
The best way to learn how and why the horse's muzzle is tick-tocking, or not, is to watch several advanced horse and rider pairs.
