I use a the one rein stop, however only from a walk or trot, the version I use is not really a spiral down. It's a fast whipping of the head around (at fast speeds you can pull a horse over, it's how stunt horses fall in movies) however it's also using leg to move the hind quarters over.
I will use it to stop a buck, if a horse breaks gait, sometimes in a spook, to stop a bolt before it happens, I will also use it gently to just stop a very green horse or a horse with a hard mouth.
It's something that needs to be taught to both horse and rider. It also needs to be practiced (one each side every time I ride) to maintain it's effectiveness, it can be used with a bit, bitless bridle or even a halter so long as the horse understands how to yield to pressure. (Stiff/untrained horses generally will spin rather then stop).
The you can also pull the bit through the horses mouth, rending the stop ineffective. So a cheeked bit (fm, big d ring, English Tom Thumb) or chin strap should be used.
The pulley rein can be very harsh and I would use it if needed only on a bolting horse, where I missed the window to use a one rein stop, and I could not circle the horse
The idea of the pulley rein is to tighten and brace against the neck with on hand (to keep the horse head straight and create leverage) and pull up with the other rein. I've never needed to use it but I have (gently) practiced to make sure both my horse and I understand it.
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