I have experienced that the 'stop, drop the shoulder , and spin . . . aka: stop,drop and spin, used by horses to do an instant 180 degree turn AND unseat their rider , is much harder to counteract when you are dealing with a 'stiff' or 'boardy' horse, such as many gaited breeds are.
I think the reason their spin is SO hard to stay on through is that they swing like a gate swings; from the hinge. That means , in this case, the 'hinge' is the horse's hind end, as far back as possible. The horse is so stiff that the entire rest of the body swings as if it were an entire 'board'. And since since the horse prefaces this swing by sort of 'bouncing ' off the front legs and dropping the front, inside shoulder, YOU are propped forward on that board, before it starts swinging.
It's as if you were astraddle a gate, you somehow got pushed forward so your body slopped forward, just as the gate started to quickly swing open. you are going to find it VERY hard to stay on that swinging gate.
Whereas, a horse that PIVOTS around the middle, such as a good cutting horse, will have the center of it's turn exactly under the rider. So, you are like in the 'eye of the tornado' , and you stay solidly in the saddle as the horse pivots around.
again, I say, ask me how I know? well , it's that sort of gate on hinge swing, that ends up dumping me in the dust, 5 times at least.
that's how I know.