My first guess is that it is a cavalry saddle that has been modified for a rider who is missing their right leg from the hip down.
You've pretty much guessed it. This was my great-grand-uncle Abe Bonnifield's saddle. He was born without legs. To pay for schooling, he joined a circus and learned to trick ride. He fought in the Civil War for the South. He was mainly a messenger. He weighed around 70 pounds and could hang off of his saddle for at least an hour, which let him ride through enemy lines making the enemy think it was just a runaway horse. Here's a link to more info on him and his photo.