My horse is out on pasture vacation right now so I can’t go ride and test this out. But I think at the walk, the ribcage swings away when the hind foot comes up. As your inside leg swings under the horse, the inside back leg is moving forward. go for a trail ride behind a friend and watch how the horse and rider move together. Or go for a trail ride behind a friend that is hand walking a bareback horse so you can watch how the body moves. Very interesting if the horse has a lot of swing! Or watch the great computer simulations of how a horse moves that are out there. edited to add: which I can’t find now of course.
It’s true that when I returned to riding I could feel going short, crocked hocks, tight stifle, etc. in the older lesson horses but was never allowed to comment on it because clearly how could I tell since my riding was so deteriorated? I had only ridden a sound younger horse as a kid, so any disruption of symmetry was very obvious. If you grew up riding lesson horses you might take a certain amount of asymmetry as normal.
I’ve never had any trouble telling the canter lead, or a disunited canter. That gait just seems so obvious as it is not symmetrical. I almost always pick up the right posting trot now, but I am not sure how that happened. Certainly on any circle or bend, the wrong diagonal just feels . . . wrong. As far as telling which leg moves, I feel like my trot motion is more up and down (especially when posting) than side to side, but because the gait is diagonal, you always know wht the hind is doing by looking at the front.