Yes, horses can 'twist' in that they rotate their vertebrae around their own axis!
The really great thing is that they actually do that with every step they take: when a horse lifts his right hindleg, his right side of the pelvis sags down while the leg is in the air. The pelvis is twisted around it's axel, with the lowest point being the right half of the pelvis. The great thing is that this twist then is followd by all his vertebrae, up to his neck: as the right side of all the back-vertebrae sags down, the ribcage and belly are swung to the left, underneath the spine - clearing more space for that right hindfoot to grab forwards and to land.
You can exaggerate that twist actually by stepping under and the shoulder-in: as the horse is now not only stepping forwards with his right hindleg, but also more to the left underneath his navel, the right side of the pelvis is lowered even more in order to reach that spot, the belly and ribs swing even more to the left and with that they lift the left shoulderblade up: collection!
It's exactly what you do with your own body when you let your pelvis face forwards while rotating your shoulders in a horizontal plane. Only the horse starts the swing with the hips, not with the shoulders. I think your own stretch would also be even more intense if you don't just rotate your shoulders to the right, but at the same time rotate your pelvis to the left. That gets all the vertebrae going!
_________________
New horse book: Mandala horses! Never stop making mistakes!
Natural Dressage