Yes, certainly with some horses.
My mare is a jackrabbit/arabian cross BUT she takes you with her. When most horses exit stage right, they drop the left shoulder. Miss Mare picks up the left shoulder and throws herself (and you) to the right, which is easy as pie to sit. It also looks impressive as all get out to people on the ground … the big suspended gaits, the flagging tail, and 90+ degree turns at speed and you’re up there cool as a cucumber, barely moving in the saddle.
She’s also the type that, if you do start to come off, she’ll try to step under you or stop all together until you get yourself back in balance. For these reasons, I put children & husbands on her because she’s a complete babysitter even when she’s having a flighty day. She totally works at keeping her rider up there.
I’ve only come off this mare maybe 5 times in the almost 20 years I’ve had her and every time, she’s reacted the same way … by coming over to me, sniffing me from head to toes and then not putting a foot wrong afterwards, sometimes for weeks afterwards. No matter how wound up & bouncy she was when I came off, once I was back in the saddle, she was like a 30 year old trail pony … head down, flat gaits, constantly checking back on you to make sure you’re okay. I even came off of her once on a hunt, cantering with a group and she turned around, came back to where I was sitting in a brush pile and checked me over, head to tail. Rest of the hunt, she was the opposite of her normal, high on life & cross country self.
You can tell, she’s totally sorry about loosing you and she’s not going to do it again.