Canter
The best gait to feel the circular movement of a horse's back is canter. When pelvis matches horse's back in canter it starts moving in a sort of backward circle. The forward downward momentum of a canter stride is matched by upward and backward motion to complete the circle and return to neutral position upon landing into the next stride. The moment of suspension though feels weightless will not create any force that separates rider's seatbones from the saddle. The rider lands with the horse not after. Why is so many riders appear to be struggling to match the stride and look like they leave the saddle and then hit it again and again? Because during forward/downward part of a canter stride such rider will be either too passive and slide forward/downward or even worse the rider will add her/his own force to it and slam the horse with the seat thinking she/he is "driving" the horse. The correct dynamic of the canter is such that by the third beat the horse is already coming up in the withers and if the rider is not matching that she/he will slam the horse behind the withers, dropping it on the forehand and then will be late to come back with the horse to start the next stride.
The rider who is very stiff in a lower back will look like their primary hip movement is backwards and up behind leaving the saddle every stride. The ones who "drive" with their seat lean back and prevent themselves from leaving the saddle but end up permanently behind the horse's motion. Many horses who have sensitive backs and/or flamboyant personalities will buck or run in either cases. The first group of riders feel to them like the buck is not just allowed but encouraged and the second group of riders make these horses feel like they are squished down like a bug and they rebel against it.
In order to change the canter from working to collected or extended a rider must engage the horse and use the seat to "tell" the horse what canter she/he wants. For example, a working canter has a circle shape, extended canter creates more of an oval shape laying on it's side. Collected canter requires an oval standing upward.

WALK
The walk is the second best gait to feel the complexity of the horse's back movement. Even though it does appear while watching a rider on a horse that hips move back and forth while horse is walking it is not so linear. There is also substantial sideways motion in horse's backs in walk. And if you watch very carefully you will see many riders move sideways so much that they collapse left and right with every stride. They think they follow their horse's motion in reality they are exaggerating the lateral motion in the walk inviting the horse to dissipate it's energy sideways instead of engaging it forward.
Others will appear slightly leaning forward in their upper body and their hips move rather backwards every stride behind their upper body.
The ones that sit vertical and let their hips swing forward in front of their upper body every stride are more correct. However, it is very important that the swing forward is aided by core muscles helping the pelvis to come up at the most forward point of the swing rather than add force from behind and push the hips into the horse's withers. The feel of the pelvis lifting up at the pubic ramus is similar to an image of "pulling up on your bikini line" (Frances Carbonnel). The pull up is gentle and happens rhythmically with every stride. It is not equal on both sides, it is slightly more on the side of the front leg swinging forward. It alternates with every front leg giving a horse permission and support in swinging it's shoulder forward.
It is also beneficial to imagine that each seatbone describes backward circle in alternate mode - left seatbone - right seatbone, etc.
When a rider can feel that much in the movement it is quite easy to know when the horse is uneven, pushing to one side or stiff. Riders who are busy to push with their seats will miss subtle changes happening in their horse's backs.

Trot Sitting trot is most difficult gait to feel the circular movement. The alternating diagonals create a sense of up and down movement like pistons in a car engine. However, again these movements are rather alternate circles than straight up and downs. If a rider is not careful and tenses up particularly in lower back the seatbones and hips will not be able to move in alternate mode but rather bounce up/down as a whole thing. If legs clamp to the horse's sides the rider's seatbones will lose their connection and bounce without control. The pelvis should be allowed to match horse's movement by describing not very obvious but clearly felt alternate circles which are guided by core muscles. The lower back stays supple and the rider appears to be gently bouncing up and down without ever leaving the saddle or disturbing the horse. The pelvis movement is not very obvious and you have to know exactly where to look to recognize it.
In rising trot the movement of the pelvis is still circular because the pelvis rises up and goes forward on top of the thigh bones that act like radius of circles with the knees being the centers. Riders who perceive rising trot as only up/down movement push into stirrups too much and end up behind the movement.
The variation of a trot to clearly feel backward circles with seatbones is half-steps/piaffe. In this movement the forward progression is minimum but the horse must come through his back and topline no less than in extended trot. Because there is very little forward progression the feel of an alternate backward moving circles is exadurated and becomes very obvious. If the horse feels too linear like his energy is pushing forward instead of upward and back he is falling on the forehand and is not using himself correctly.
When a horse becomes more through and responds to half-halts better it starts to swing more and engage more because it learns how to recycle his own energy and send it up and backwards to create expression and lift in his trot. A rider who cannot control the pelvis position/movement will not be able to encourage her/his horse to find that lift if the horse does not have a natural aptitude for it. With a talented horse such rider will not be able to bring the best and most expressive movement the horse is capable of.

Happy riding...