Question: I am a Professor at a college in the US where we wear regalia to the student graduations. I have a PhD, but am about to complete another doctoral degree from a different university, neither of which are where I teach. Do I have to choose one gown to wear to graduation or is there a specific way to represent both degrees? I have yet to find any guidance on this!

Answer: 
Yes, you can wear whichever you choose, but the intercollegate rules laid down in the late 19th century suggested (but did not require) that you should wear the regalia of your most recent terminal degree.

On the other hand, I suppose you should probably wait until you earn your second degree before wearing it as part of your current faculty function. If you choose otherwise, as a degree-in-progess individual, your tassel would be worn on the right since you have not graduated yet.

I imagine you'd want to wear the gown of your degree granting institution at your own commencement as a student, and I'd check with your professors before doing otherwise.

The most important point is that you should not attempt to blend both styles into a uniform resembling a clown costume. For example, if you have earned hoods of two different colors, you should pick one color and its associated regalia and not attempt to include the other.

<http://intercollegiate-registry.org/revised-intercollegiate-code/>
@Ragaroni's answer seems good for the US. Here in England-and-Wales, there are no intercollegiate rules and it would be a question for the internal regulations of the university hosting the graduation ceremony.