Your horse likes to be rubbed, where your dog likes to be patted. Your horse likes to see everything around it, where your dog likes to sleep under the table. We have to understand first, that our horse doesn't interpret predators. This makes it our job to understand, and our job to make it work. Until you start getting down to the nitty grity of communication, you will not run into anything that the horse cannot do.
All horses can run, jump, change leads, side pass, and backup.
The problem is that not all riders can communicate this in a mannor that the horse will understand. A great number of problems start by people believing in what has been dubbed the three lies. You saddle him up and get on. Give him a kick to go, and pull on the reins to stop. Now go ahead and ride.