Cool. Where i use to trail ride guide in the mountains had 6 jack russels. THey would occaisonaly come riding with us. Either on the back of the horse like that. Or if front standing on the saddle horn /against us. Hikers were always doing double takes when they passed us.
One of them would also ride in the saddle bags or inside my slicker if it was cold.
Friends of mine who happen to breed Maine Coon Cats, have one called "Hollywood Jake" that used to ride in parades (wearing hats) and trail ride for hours on end. He's 13 now and too old to ride, but he still goes camping with them.
He's been on several TV shows, in TV and print ad commercials, etc
They've had several others that would ride a little bit, but Jake just loved it. They would attach a thick rug to the back of the saddle and he'd just sit there. He learned to balance and could trot and canter. If he got unsteady, he'd just jump off. They also got a backpack and he'd ride in that. It was so cute how he'd hang on the rails with his front legs and just watch the scenery go by.