I’ve ridden H/J/Eq since I was a kid. In college I started riding western, mainly HUS and horsemanship. Needless to say, I’m right handed and ride with my left hand. I find that since I typically ride english, I’m less likely to grab at the horses face with my left hand. Learning to not have much connection with the horse’s face was seriously the hardest things about switching disciplines, so being somewhat less responsive with that hand was helpful. I was taught to hold your rein hand in front of your belly-button and then pretend there’s a box around it and you may move your hand in said respectable box.
Also, since I’m right-handed, and therefore stronger with the right hand, it took less energy to remember to keep my right hand closed (softly…unless I was having a hard time on the rail or in my patthern, then it was clenched so tight) since it is my dominant hand.
What do you do with the other hand? lol I guess whatever you want at home. I usually held a rock when schooling at home for shows. Reminded me to keep my hand closed and I had to make a conscious effort to hold my hand like I was holding a beloved ice cream cone. Or you could hold a real one