It rears (raises its front legs), or it shies (tosses its head). This sort of 'shy' is not used in any other context. In fact, sometimes people unknowingly rewrite 'Jimmy Crack Corn' (as here: Blue Tail Fly - Wikipedia, the free encyclopedia ) to replace it with the rather banal 'the pony being very shy' instead of 'the pony being apt to shy'. Come to think of it, the 'apt to shy' version may be the arriviste - it's just the version I heard first.
In show jumping, a horse that does this and stops before jumping a fence is said to 'refuse'. It's intransitive: Her horse jumped all the fences until the last. Then it refused.