Saddleseat riding style is a colonial era American riding style, loosely based on classical riding.
In general, the riders sit on the rear portion of the saddle, back beyond the "rider's groove", have feet on the dashboard, unnaturally elevate the horse's head, bringing his neck beyond the vertical.
This "caricature" type riding was based on pre-Caprilli thought that weighting the back end of the horse with the rider, would lift the front end, resulting in higher leg action.
This picture shows the rider sitting back on the cantle of the saddle, on the horse's loins, and with feet "on the dashboard", braced against the stirrups:
The horse is basically carrying the rider's whole weight on his mouth which may have a broken jointed snaffle in it, and be clamped down by a tight noseband.
Saddleseat riding was (and sometimes still is) used on gaited horses. It was thought that sitting further back on the horse lightened the front end. This is incorrect, proven over 100 years ago by Caprilli and his use of the forward seat. Biomechanically, it cannot happen; the horse does not operate like a motor boat, a bike, or a rocking horse. One cannot weight the hindquarters by sitting on the loins, to lift the front end.
We understand that some trainers and their students believe this to be correct. We need to improve knowledge and education of these issues by studying biomechanics and looking to the gaited horse experts who have studied biomechanics. No sense in perpetuating incorrect information or riding styles. If you have a trainer that uses this riding style, he is not keeping up with state of the art knowledge of equine and riding! Find a good trainer who is better educated and understands biomechanics and how horses move.
Saddleseat riders carry their hands higher to encourage a higher headset, which in turn hollows the horse. Saddleseat horses sometimes have their feet manipulated with weights or action devices to get more "flash" from the front end.
This seat is still used in Iceland and Europe, particularly in shows / evaluations / competitions.
This rider has her legs more under her than the first picture, but she is still sitting on the horse's loins (the weakest part of the horse's back).
For a natural horse and for long-term soundness, and in consideration of the horse, a balanced, centered seat is recommended. Allow the horse to use his body more naturally; to do the gait to which he is conformationally inclined.