I saw a clip from an upcoming rodeo type program, and one of the riders had his horse do what might be a common move for equestrian events, but I have never seen it. The horse was trotting with his front legs extended normally, but his back legs were kinda scrunched up under him, almost like he was scooting along on his butt. My question is: how in the world do you train a horse to do this? With a dog, you push his ass down, move him forward, then give him a treat, or such like until he gets it. That wouldn’t work with a horse, I don’t think.
You’ll probably get better answers with a link to the clip in question–just offhand I can come up with two or three possible legit horse moves that might fit your description. It would depend a lot on the tack used and what kind of horse it is.
Yes, there are several things it could be. A clip or a bit more description (what part of the world, what event, etc) would help.
I’ll pay more attention next time I see the promo. I thought it might be a common horse trick.
From watching dressage shows, it’s clear that horses can be trained to do some downright fancy choreography. (See any video of the Lippizaners, for example.) Horses may be dumb (and many horse people would dispute that), but they are definitely highly programmable.
It’s from a promo for “The Last Cowboy”:
Is the behavior you’re asking about shown in that clip? Because the closest I can see is the way roping horses slide around on their butts a lot–they learn to do that because when you rope a steer and it hits the end of the rope if the horse is standing up normally the steer can yank the saddle right off or yank the horse over and make it fall. To keep this from happening, the horse is trained to face the steer once it’s roped and stay that way and it hunkers its butt down like that to lower the center of gravity and get its hooves spread out to create the greatest amount of resistance to the yanking steer on the other end of the rope. The horse keeps the steer busy and controlled with a tight rope until the rider jumps off and gets it hogtied.
So it was part of roping, not just running in a funny way 
Yeah, pretty much, but you do have to train the horse to be a butt slider as that doesn’t come naturally much, aside from going down a really steep slope. One method they used to use was to put shoes on the horse’s back feet but leave the front hooves barefoot–the tenderness of the front hooves would encourage the horse to keep his weight over his rear feets. After it becomes second nature you’d shoe the front feet too.
Most of that is Reining, of which the sliding stop and the spin are a big part (and they are indeed taken from things a good cowhorse does naturally). Those moves require the horse to really “sit” over his hind end, to carry his weight and free up the front end. There isn’t really any true forward movement. The sliding stop is moving only because the horse has been running flat out and inertia carries him forward in the sand.