You can palpate is hip to see if he is sore... or anywhere in the left hind for that matter. The fact that he avoids picking up the canter by rushing says he doesnt want to strike off. For the right lead, the horse strikes off with the hind left.
When I was first started dressage work with Bailey he got sore in his left hip. It didn't show up intil I cantered right. He either rushed as your horse is doing, and when I gave a stronger canter aid he would jump into it with a bit of a hop/buck and then the canter felt more like a pogo stick.
As JustdressageIt said, the canter is assymetrical, where are the other gates each side is moving relatively in the same footfall. If there is a problem it will show up here.
Also the possibility that you were riding a little crooked that day... :) Drop your stirrups and do some no stirrup work at the walk and trot (loosen your hips and find your center) then try and ask again.
Ask from the walk and see if that helps too, where you can be more organized when you ask.
Id suggest an equine massage therapist before the chiro, when Bailey was sore I had her massage him and it worked wonders.
"The white horse moved like a dancer, which is not surprising: a horse is a beautiful animal, but it is perhaps most remarkable because it moves as if it always hears music."