Given that I am crooked, I may not be the best judge of side-side motion
. But I strongly suspect a horse built like Bandit doesn't generate strong side to side motion because it is extremely obvious on Cowboy, even to me. This is Bandit, shaggy still and looking around, but the best recent photo I could find of his build:
I still remember the first time I tossed a leg over him, thinking, "Where is my horse?" Mia was slender, but Bandit far more so!
I also suspect his background and why he was bred plays into it. He was bred for relays races, with 10-15 mile legs. The F-4 was my first love in planes, but I only did one tour in it. Much later, I found myself in the EF-111. I came to love it almost as much as the F-4 (Internet picture of the EF):
The EF-111 didn't have big enough engines to accelerate quickly. But she was streamlined as all get out and could, with time, haul butt at impressive speeds.
Bandit is more like an EF-111 than an F-4. Not a lot of thrust, but I'm told he was quite fast covering 10+ miles. If you are right,
, then part of why he doesn't create sideways motion is HOW he moves. Not a ton of thrust, although he can build up good speed. Low, flat strides. He gives an elevated, floating trot once in a while, but his normal stride is meant to cover the most ground with the least energy - as makes sense for a horse who grew up carrying 200-300 lbs on 10+ mile legs of relay races!
A QH would generate a lot more thrust. In airplane terms, the hind end is the engine and the front legs the wings. Larger thrust generated at wider distances from centerline would create more lateral motion - particularly if the rider values an elevated trot, so the front legs engage to lift the withers. Bandit's front legs mostly are used to keep his body level. Elevated strides are a waste of energy crossing the desert.
"When the horse pushes off that right hind, the hip that is bearing the most weight is the highest of the two hips, then it is lowest (at the point where that rear leg is thrust furthest backward.)
When I compare a horse to a plane, I'm being very simplistic. An airplanes engine is bolted in and only generates thrust directly behind it. The leg of a horse, like a humans, can pivot and create sideways thrust as well. We do that all the time when walking and running. A close-up of Bandit's rump at a trot. I'll delete this rather inelegant picture in a week or so, but...or "butt":
His hind legs still are close to centerline because he angles them in. I have no way of knowing how he angles any thrust. It looks like one side of his hips may be a very small bit higher as you mentioned and for the reason you gave, tinyliny. But there is no reason this small elevation must cause his entire body to rotate. Our hips can pivot easily without affecting our shoulders. I've never seen a study, but I'd bet money the pelvis of a horse can rotate without affecting the rest of his back.
In the end, so to speak, I guess it boils down to this: any lateral motion Bandit generates is too small for me to care about. If I cannot feel it then I cannot compensate for it. Nor need to. On Cowboy, ANYONE will try to compensate SOMEHOW for the rocking side to side, while probably wanting him to transition to his very smooth, easy to ride canter. And a lot of horses will fall somewhere in between.
And we haven't even started discussing how a horse trots on UNEVEN ground, which is 95% of Bandit's trotting. Much of our trotting (and cantering) is done is washes. All the theory goes to heck when the horse is trotting in one of our favorite spots:
As for diagonals when trotting in the wild...I don't think it matters. Jogging in the desert is a motion utterly unlike jogging on a track. At long distance - an endurance racer - it may matter a lot. How much exercise each leg is getting on uneven terrain is impossible to calculate, so I figure the horse will have to adjust based on how he feels.