To add to what I just wrote above, look at the legs when you're going to the right: they're landing in the correct order for lope. Three beats: hind left first, then hind right and front left at the same time, then front right.
Now look at your horse's feet when he's going to the left: despite the head motion, he's just doing the two-beat steps of the jog/trot. Hind left and front right together, then hind right and front left together. For lope that direction, it *should* be the three beats as above, but with: hind right first, then hind left and front right together, then front left.