I don't think I've ever seen T play with his ring, but I will notice myself doing spinning my wedding band/engagement ring, taking them off, putting them on other fingers, putting them perpendicular to one another at random times, which are sometimes related to my relationship or marital relationships and sometimes completely unrelated. I don't do it very often and I have no idea why I do this sometimes as opposed to other similarly distracting things (pen clicking, foot tapping, etc.).
My T will shake his leg or drum with his fingers sometimes and as we sit next to one another now, it is sometimes really noticeable and I tend to read into it that he is impatient or bored or frustrated, but most of the time, I think it's just him being a little fidgety. He is ADD (I mean literally, from his own mouth) and can be a bit distracted at times. He's really into music too (has played drums, guitar, bass guitar, lead guitar, etc.), so for all I know, he's playing something in his head while we talk.
I agree with Cat, too, that even when we think we would have noticed something, it's possible that we blocked it out and didn't. In my case, this tends to happen a lot...but, I can barely look at the guy, so I might not be a good example.