The capriole is the most difficult of the airs above the ground - at the Spanish Riding School, only the very best horses perform it and then often without a rider.
The horse leaps off the ground and then kicks out with its back legs...if done correctly, at about head height. It's thus a maneuver used to attack an opponent approaching from behind.
The last of the Ludwig Koch pictures I found. Note that the horse's ears are pointed backwards, as one would expect if somebody was behind him.