A reining horse has to accept the rider asking it to give its face. You cannot teach much on a loose rein. The horse has to tuck its chin and 'step up into' the bridle when asked instead of backing away from the bit.
I would guess that these horses have not received enough preparation to spin correctly. The proper preparation includes taking the horse into small circles with its head down and tucked, its nose to the inside and the circle becoming ever smaller. This teaches the horse to NOT resist the bit, to move forward into the bit and to cross over as the circle tightens.
A rider may have to roll a spur (not jab or poke) into the horse's ribs when the circle first tightens, but one should never ask for more speed or a smaller circle until the horse willingly stays on the size circle that it is doing without a spur. As circle become smaller and smaller without spur pressure, the horse is getting ready for more speed and for the circle to tighten up into a spin.
While the horse is walking up into the bit and doing the small circles, the rider keeps BOTH legs on the horse to keep good forward movement. This is important, because the signal for the horse to turn around is going to be the rider taking that inside leg off of him.
The very first turn-arounds should be initiated by:
1) the rider dropping the inside leg off of the horse
2) tightening the outside rein slightly to halt walking forward
3) rolling the outside spur into the horse's ribs
4) smooching (this will later replace the spur altogether
If a horse is properly prepared for this maneuver, they actually try to stop and turn around before you want them to. I never let this happen and always trot a horse straight forward if they try to stall out and initiate a turn-around on their own. Then I go back o walking the small circles.
The first turn-arounds should only be a step or two and then the rider should walk or trot the horse forward out of it. You always stop the turn-around before the horse loses form or before you have to push it harder.
If you always smooch when you ask, pretty quickly this becomes the que and you avoid the resentment and switchey tail that comes from a horse associating a turn-around with a spur.
The turn-around is all about shoulder control. You do not 'hold' the horse's back end still with the outside leg or spur. The pivot foot takes care of itself when the shoulders are moving freely.
Hope this helps.