If he's a riding school horse, then you'd still need to check with the instructors.. just because you ride him the majority of the time, doesn't mean you can make such decisions without the permission of his actual owner. Maybe you are the actual owner, but it seems like you're just his main rider.. so I'd check it out with the barn owner. :)
Also, if he starts backing up while you're riding him, I'd keep him backing up until you feel like he's learnt a lesson. That's what my instructor has always taught me to do, for many situations.. ie: he gets a little bit wound up and takes off with you on his back, get off, put a lunge line on him and canter him until he can pick up a nice, collected canter. :)
Forever loved, never forgotten; my beautiful Indie. <3 Hoofprints on my heart.