Reading this page,
I realized that sidesaddle IS saddle seat. Hunt seat sidesaddle is a retrofit.
Back in the 19th century, Frederico Caprilli had not been born and there was no forward seat. What we now call saddle seat was just plain riding.
Many years ago, I had the good fortune to show sidesaddle.
This is hunt seat sidesaddle. Saddle seat tack/attire would have a different bridle, a derby instead of a bowler, shorter boots, and longer jacket & skirt. The horse is George. Bless his stodgy self.
To maintain a hunt seat position, I had to fight to keep my weight centered in the saddle. The whole construction invited me to put my butt on the cantle. Exactly were my saddle seat instructors keep telling me to sit. Note, this does not mean “in the back seat” nor behind the motion but physically toward the rear of the saddle. The rider is still in balance, just farther back.
When riding in a side-saddle, my hands had to be up because my knee was in the way. I had to work to keep my hands down to follow the mouth of a Thoroughbred going in a long, low, hunter frame.
I mentally unwrap my leg from around the pommel, drop it down alongside the horse, and voila – saddle seat.