Wiktionary
n. 1 A slightly concave seat (used in a Windsor chair) that sometimes has a thick ridge in front 2 A style of horseback riding, designed in the United States to show off a horse's gaits, especially the trot.
Wikipedia
The goal of the saddle seat riding style is to show off the horse's extravagant gaits, particularly the trot. In the United States, there sometimes is confusion between saddle seat and hunt seat disciplines among individuals who are neither familiar with different styles of English saddle nor the substantial differences in rider position and attire between the disciplines.
Usage examples of "saddle seat".
His cloak was stuffed underneath to help balance the chest on the rounded saddle seat.
Pivoting, he swung it onto the top of the stall's partition wall and smoothly met his father's look across the saddle seat.
Improvised strap rigs had been sewn together to form a makeshift saddle seat for the back of the men so that the children could be carried like packs.
The Indian purred as it idled, with hardly a vibration through its leather saddle seat as Wolf parked his bony posterior on it.
Kubera, who had carried Sam to the pens, strapped him into the saddle seat, still unconscious.