I would much rather be licked ("eaten out" is just such a creepy term, to me) than have penetrative sex while on my period. If I've just showered, and am either wearing a tampon or a menstrual cup or am at the tail end of my period, then it's all nice and clean, and what's the big deal? I've also had to talk my partner into being okay with me going down on her under those circumstances.
I used to be the other way around, back when I hadn't yet realized I was gay--I'd be okay with penetration, but not oral. I think knowing what I feel comfortable doing to another woman makes me more open to allowing another woman to do those things to me.