Hinge with part perpendicular to other
It's possible I'm picturing wrong what you're trying to do, but perhaps a cabinet hinge like this?



I don't know if that hinge will be strong or robust enough for your application.  Cabinet hinges don't really have to endure much force.  You might be better served with what is apparently (aptly) called a "center-mount hinge":



Which sandwiches whatever is supposed to pivot.