Rearing is an evasion. It is usually brought on by ill fitting equipment OR by asking the horse for more than he is trained to do. The horse feels trapped and rears. If the evasion works, he will do it again.
Most rearing of a mental nature (tack fits well, bit is OK and so forth) is a lack of foundation training. The horse may mask this until one day it comes out and he has no real idea what you want.. and so he evades by rearing. From there he may default to rearing whenever asked to do something he is uncomfortable with.
From your description of a Hard Mouth and him rearing when asked to slow down or stop, it sounds to me like his foundation was missing a few footings. He was likey unbalanced when asked to transition down and that lack of balance scared him.. he over corrected and reared.
Horses do not really get hard mouths (they learn to evade the pressure by boring into the bit) and most horses are not ever taught how to slow down and transition from a faster gait to a slower one (or a stop) without losing baloance and ending up on their forehand and "falling apart." If your horse felt like that (or like he was going to fall) he might rear.
If rearing becomes habitual, it can be very difficult to fix and the horse may never be reliable. OTOH if rearing is evasion, the best thing to do is go back and plug the holes in the horse's foundation training before asking for more so the horse has no reason to rear.
There is something about the outside of a horse that is good for the inside of a man. ~Winston Churchill
) Dinosaur Horse Trainer