Alright folks, time for the million dollar question...who knows what these are for?
If you ride a western saddle, there is most likely one of each side of your pommel. Some endurance saddles will have them as well.
These pieces of leather are called latigo keepers, off side cinch holders, and also flank cinch holders. The piece of the horse and rider's left hand side is used to hold the extra latigo after cinching up, like below.
Now, the off side, or flank cinch holder isn't as commonly used. Nothing irks me more than to pull a saddle off the rack and have the cinches drag on the ground. The piece of leather on the horse and rider's right, also known as the off side, is so valuable! With it, you can connect your flank cinch to your front cinch and secure it to the holder, keeping it off the ground, and, most importantly, not dragging or being flung over the horse's back in a messy and slightly rude fashion. Below is a short and simple video on how to use the flank cinch holder.
Hope this helps keep your tack room and saddle a little bit more clean and organized!