The canter that has the most equal spacing of its four phases (three are heard, the fourth is the silent suspension moment) is MEDIUM CANTER.
Changes can be performed in collected canter, where the diagonal pair lands sooner than in medium canter and suspension time is absorbed into contact time of all the legs. Below is a movie of one flying change from left canter to right canter, performed on a loose rein. Vulkan (24 year old Württemberger gelding) begins the change in time one of collected left canter (right hind grounded) and finishes the change with the left hind grounded. The hind leg swap in collected canter occurs while the inside fore of the left canter is still grounded. This instant is so brief that you will have to step through the movie frame by frame to see it. The movie ends at the landing of time two (diagonal pair) of the new lead (right canter).
If the movie does not play evenly on your computer, let it load and step through the frames with your keyboard or with the QuickTime controller.
The canter for changes has a specific aid: the rider's legs are kept exactly parallel until the canter is released onto the new lead. So the left canter lead aid uses the left leg at the girth ( activity timed with each stride keeps the canter lead), right leg slightly behind the girth: then for the set up of changes, the legs are parallel (number of strides determined by the rider), then the change is released by a slight repositioning of the legs. In the movie above, my right leg is repositioned for another change at the end (last change not shown, as the action moved off camera). The loose rein is used to emphasize the riding of changes from the rider's lower body.
Dressage flying changes are often treated as mysterious. But they are based on the changeover from one half of an asymmetric gait to the other via a specific leg swap sequence.
Below is a movie of Rio Sereno (15 year old Canadian Hunter Rhinelander cross) performing two canter counter changes of hand. From behind you can see the hind legs swap position in the air. Sorry the movie image of the horse is so small: I will get another movie made soon (need a videographer). Meanwhile, look at the still frames from the first change for the moment of leg swap in the air. At the moment of the change, the legs are in a different position (see three legs evenly lifted in still frame 2 and the left hind just about toe-down) than when Rio is in true canter. The flying change is a sequence entity with its own distinctive leg motion timing series different from but related to canter stride sequences.
It is worth noting that the stride rate of canter is modified from about 105 strides per minute for extended canter to about 65 strides per minute in pirouettes. The canter that prepares the horse for changes has the hind legs more under the body and the "canter for changes" looks slightly different to the eye for that reason.
With this information, we can look at the movie below and understand that at the moment of the change, the horse is in a lateral position, just as he is in the changeover from one half of the walk to the other. Take a look at the walk frame HERE FRAME SIX if you need to refresh your memory about the leg positions at the "vee" of the walk.
The horse in this movie is doing two tempi changes.
This next page has the still images from the movie above with a commentary on the changes shown.