Often, when clinicians explain why teach HQ yields to your horse, they emphasize the control aspects:
"The horse's power comes from behind."
"Control the hind end, control the whole horse."
"The brief moment that the horse's hind legs are crossed their balance is compromised and are being able to get them in that position reinforces you as the leader."
All of those statements are true, and important reasons to have good HQ control, but I feel that not enough focus has been put on the psychological element of HQ yields, and how it can help us build trust and confidence, more than just overpowering dominance.
To understand the instincts and psychology at work we have to look at the times a horse would yield its hind end in the wild. There are basically only two circumstances a horse would cross its hind legs and make a pivoting turn like we ask for in a HQ yield. First would be when acquiescing to a more dominant horse, which as has already been pointed out, vital to getting our horses to view us as alpha in the herd. The second instance a horse will make a pivoting HQ yield in the wild is the one I would like to focus on. This would happen after a horse has been spooked and is trying to assess the danger of the thing that scared it.
Imagine our mustang standing on an open plain of sage, suddenly a rabbit moves startling our mustang. The horse jumps forward a few steps, then sharply pivots around to have a better look at what scared it. It takes a moment to think about the fact that the rabbit is no real threat and everyone goes on about their business. Different horses will spook a differing number of times before they realize spooking at cottontails is a waste of time and energy, and rabbits can't scare them any more. What is important to realize from this scenario is that the HQ yield is what triggers a thought process. Just after the horse turns to face its fears is when it takes a moment to think, "Hmm, was that really going to eat me?"
One of the most important things to remember about a HQ yield is that it triggers a processing mode in the horse's mind. If the horse is yielding from a dominant horse they will be thinking: "Yep, she is still the boss today." If the horse is turning and yielding when it is even a little scared the following moments will either reinforce the fear, teaching the horse to run farther, fight harder, or will help the horse overcome those fears and view a particular stimulus as no threat.
This is the reason why many trainers will have a horse go back and forth, doing HQ yields several times near a horse trailer, before ever pointing the horse's nose into the open door of the trailer. Or why, when riding, trainers practice bending and yielding young horses a lot in the first rides. All of these yields will cause the horse to be thinking more and reducing their perception of danger.
While HQ yields are very important for establishing and maintaining respect, it is also a valuable tool for building trust and confidence, as long as the horse is given the time it needs to think and process the situation as not dangerous.