If the horses DO do more than one event, it's usually in the highschool rodeos and local rodeos, not in the bigger rodeos.

I've seen calf horses that were breakaway horses (that's a given, of course) that were also good barrel horses.

I've seen head horses and heel horses that were decent/good barrel horses as well.

My dear friend has an absolutely AMAZING Special Effort gelding that has won a lot of money both on the track and in barrels... He's stout enough and level-headed enough, he could make a very good head horse. So she's started swinging a rope on him. It will be interesting to see how well he does.

A great deal of intelligence can be invested in ignorance when the need for illusion is deep.
Saul Bellow (1915 - 2005)