As I think the advice here makes clear, there is no "correct" answer.
As mentioned, some workplaces will directly forbid employees dating each other, especially if one is a supervisor of the other. Any workplace that is big enough has ended up with a problem with two employees dating, then breaking up badly, and ending up with complaints about favoratism / punishment, or requests to "keep separated", or worst-case scenario a real retraining order (which would require, somehow, that an employer keep two employees physically separate). Such rules and complaints may or may not be legal in the end, but they cause amazing pain and suffering for everyone, so large employers can simply forbid it as a matter of policy.
You may ask "but I'm not working with this person ever again, so how could this be a problem?". Well, let's say you date her, and then break up badly. She is clearly angry. You then find out that you're not getting promoted to the next year. Maybe your performance was the problem. Maybe she was "connected" to someone in your department, who pushed for you to be let go. No way to tell, and could be ugly all around.
So, it depends whether such a rule exists -- and you should check at both the GME and the Professional Staff level. If not, then it depends on how much you want to "risk" the above. Note that the above outcome is rare -- most people have a relationship, break it off, and that's the end of that.
As noted, there's the rumor mill issues, etc. But that's all minor. Chances are, you'll be fine doing this. No matter what, I wouldn't ask until after the rotation is completely done.
