Judging from the departments at my undergrad university, having degrees in multiple fields or having interdisciplinary degrees is definitely valued. You can teach in multiple departments and teach different kinds of classes as well. Many universities in the humanities and social sciences encourage holistic study, and interdisciplinary degrees or degrees in more than one subject are just that - a more holistic understanding of a topic. In the Asian Studies center at my old university pretty much all of the most highly regarded professors I can think of that I had classes with were fairly interdisciplinary. But there you go - Asian history was a PART of Asian Studies, not the other way around.

It's going to depend a lot on the department, and the precise fields you're trying to combine. The tricky thing is making sure that both (or more) sides of the divide can appreciate the worth of your research - if people on hiring committees don't see the point of (or don't understand) your research and methods, then you're going to lose out to candidates whose work they do appreciate. I think the key is code shifting as much as anything else, learning to write and speak in different ways depending on your audience, and keeping a keen awareness of who that audience is.

Even then though, there is going to be a lot of variation. Some departments place a high emphasis on interdisciplinary approaches/collaboration. Some places say they do, but haven't really sorted out the structures that make it possible. Depending on how the department/school/faculty is organised, truly interdisciplinary research can fall through the gaps a bit - what they want is research that can be labelled as interdisciplinary, but doesn't actually force them to change their way of doing things. Sometimes you might even run afoul of university politics - bad blood between individuals and disciplines that you won't ever really know about unless you work there. To be fair, the same thing can happen within disciplines as well - academia is a crapshoot basically.

None of these are reasons not to do the research or degree you want to, but - as we probably all should - you need to reflect on who exactly you're aiming your research at, and how you can reach that audience.