Since I have shown and trained both reining horses and cutters, let me straighten this all out.
A spin is ALWAYS a forward movement and the horse ALWAYS has to cross the outside front leg over and across the inside front leg.
The roll-back AS A CUTTING HORSE DOES IT IT is ALWAYS a backing exercise -- never a crossing over or across. When training a cutting horse to come around, you back it up, pull it in a circle as it backs and it then 'sweeps' the ground with both front feet, staying as low in the front and in back as possible.
Let me explain why: A cutting horse never turns more than 180 to 190 degrees, even when it is fuly parallel to the cow. It backs into its turns, sweeps the ground quickly, has its hocks really flexed and low and is ready to jump forward if the cow goes that way or sweep the ground back to the original direction if the cow goes that way.
The reason you train them to do a backing up rollback is that if you let them do a 1/2 of a spin instead, they will gradually work their way forward and they cannot do that without losing points as well as being slower.
If you keep a horse in a round pen too long, you actually teach them to back away from the cow with each turn. The horse does this early in his training to 'cheat' the cow, keep a working advantage and not lose the cow. We call it 'guarding the herd'. As the horse gets closer to ready to show, you have to teach them to work straight across a pen and not back up into the herd or make a semi-circle around the herd.
Every trainer warms up their cutting horses just before the class by backing them in circles and then bringing their front ends over and around. This limbers of the horse and hopefully keeps him from driving his cows too far out. [I have seen green riders that did not get their horse to back as they did their rollbacks and they actually drove a cow all the way to the judges with the turnback riders trying to haze the cow back to the herd. It is a 'get down and laugh moment for everyone else and a red-faced moment for the green rider as he goes to or past the judges.]
There are two trains of thought on training reined cowhorses because they have to do it both ways. Some trainers like to teach a horse to cross over and spin first and teach it to back into its rollbacks after it has mastered a spin. Others swear you get it better if you start with the backing rollback.
Boyd Rice has been as successful as anyone having won the Snaffle Bit Futurity and about every big cutting event in the NCHA. He can do them both right and he teaches them simultaneously. He works on each every day and says his horses soon learn the different way he sets them up for each maneuver. When he starts working cattle, they quickly learn to back into their rollbacks to protect themselves and keep their working advantage. They stay low and rocked back on their hocks. He says they learn to stand up and cross over to spin. He just nudges them forward and then asks them to turn around.
If you watch a reining run, you will see that after a horse stops, the riders lets the horse stand up behind and get ready to spin. They will hesitate for several seconds and frequently nudge them forward.
If you watch the cutter, the horse stays down behind and has to 'sweep' the ground to get around quickly enough to stay head to head with the cow. The cow would not give him time to settle his horse and let him get him standing up behind so he could do a half a spin. The cow would be long gone and back in the herd.
I just sent a home-bred blue roan daughter of Mecom Blue to Sandy Collier in California. I raised her (with a partner that paid the stud fees) out of my Smart Little Lena daughter. She was started in cutting last year and did not do great, but we could not find a trainer that would take her that really fit her. I can't ride much any more and cannot train at all, so we though Sandy might fit her pretty well and could try her for reined cowhorse. She had only been taught to back into her turns and it has taken Sandy about two months for her to understand the difference. Her turn-arounds are still really slow and Sandy is not real impressed with her yet, but we will see where it goes. If she does not make it to the show pen, she will be the best bred best looking team penner in the world.
I hope this all makes sense.