How important is the "major" of the PhD?
It depends more on the aspect of your research. If you have a physical chemistry degree but do research more in the realm of physics, it is possible that you could be hired in physics or chemistry, even hold a joint position in both.

The limiting factor that your PhD major will be in that it is the general area where you are more trained in, that is, you can probably teach courses in your major better than those in the other department. Sometimes hiring decisions do take this into account.