How important is the "major" of the PhD?

For sake of argument, let's say I obtain a bachelor's degree in physics, a master's degree in physics, and then a PhD in physical chemistry but from a department of chemistry. Can I then go on to be a physics professor? Or would I be confined to teaching chemistry?

One can imagine equivalent situations for other fields. Say, for example, a bachelor's in computer science, master's in computer science, and then a PhD in computational physics from a department of physics. Could that individual then become a computer science professor?