Thanks to the wonderful person who showed up to my presentation at RavenCon.
Normally, you ride with a saddle blanket, which provides some padding between the horse and the leather saddle.
On occasion, you might actually have a blanket or sheet over the back. These are called "quarter sheets" and are used when exercising or training a horse in cold weather, and not planning on working the animal hard enough to keep it warm.
Quarter sheets may also be used as equine "rain slickers" - horses generally get quite miserable if they get too wet.
You might also use a quarter sheet as part of a costume or uniform.
This police horse is wearing a quarter sheet with "Police" on it as part of his uniform. (Image source: Radomil via Wikimedia Commons).