The purpose of a Hunter Under Saddle class is to showcase how easy your horse is to ride. Historically, hunters were steadfast horses that could be ridden over many different kinds of terrain comfortably. Your job, as the rider, is to show the judge that your horse can emulate those hunters.
To do this, start off with a slightly looser rein than you would ride with in an Equitation class. This shows the judge that your horse can be easily controlled. Make sure that your horse does not take advantage of this looser rein by sticking his nose of in front of him. He should stay in a relaxed frame with his nose vertical and his head down.
When you enter the ring, have your horse walking forward. He should be making your hips swing in the saddle. When the judge asks you to trot, make a clean transition. Check that you are on the correct diagonal. The judge is going to be determining which animal has the best movement. To enhance your horse's movement have him move briskly forward, almost at an extended trot. This will show him off best. Keep his frame long and low. When a walk transition is called for, make sure that he does not stall from the trot to the walk, but moves smoothly from one gait to the other.
When the canter is asked for, make a clean, swift transition onto the correct lead. Keep your horse's frame long and low, and his speed relaxed. A classical hunter looks like he can keep up this relaxed canter speed all day. You want to aim for a 12 foot stride, the classic stride length for a cantering hunter. Again, when a walk is called for, make a smooth forward moving transition.
When you reverse, make sure that your horse keeps moving forward and that your horse bends through the turn. You will repeat all three gaits again in the new direction. Remember to keep your horse moving forward with a long and low frame.
When the judge calls you to line up in the center of the ring, do so. Make sure that your horse halts square and that he doesn't fidget in his halt. Do not loosen your reins or let him stretch his neck. If the judge asks you to back up, do so with quick clean steps. Have your horse halt again without fidgeting.
Smile and have fun!

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