I can't speak for the Aggie Ring since even though I have been a lifelong diehard fan, I was never fortunate enough to earn one. However, I do know about the West Point Ring and Ags and West Pointers do share a lot of "attitudes" and "opinions", even though they are both very different schools. There may be some common history or thinking from "back in the day".
When we earned our West Point Rings, we (most of us) put them on our left hand. This was traditionally done to place it closest to the heart which was thought to be more on the left side; the same reason you place your right hand over the left side of your chest when saying the Pledge of Allegiance. Although I have heard some commentary to the contrary, I never heard anyone actually say they were trying to indicate they were married to the school. If you talk to someone from West Point in the first few years after graduation, you will get a very different feeling about the school than from an Ag. My opinion is that comes from being in that environment 24/7 for four years without much of a break.
Our Ring has a Class Crest on one side and the Academy Crest on the other. Prior to graduation, you wore the ring with the Class Crest on the side of your ring finger towards your thumb, where it was visable to you since your class meant the most to you then. After graduation, you turned the ring around where the Academy Crest was more visable to you since that was what mattered most from then on. I have also heard a similar explanation as to the top of the ring which has "West Point" and the Class Year on it. As a cadet, if you wore the ring as I described, you could read the inscription on top. After graduation, those to whom you extended your hand could read it. I'm not so sure about this, since even though the West Point Ring isn't as distinctive as an Aggie Ring, the traditional WP Ring with the overly raised stone is fairly distinctive and doesn't need much help to be identified, at least by "family".
Anyway, not being able to read my Dad's or Uncle's Aggie Rings, (worn smooth becoming part of the Greatest Generation) I can't say for sure how they wear theirs. And with "New Army", I wouldn't trust my sister, brother, or cousins to have it right
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One thing about leaving the Class Ring on the left after marriage, many West Pointers choose to do that (fewer as the years go by). They say they do it for the same reason, keeping both of them close to their heart. However, the Wedding Ring always goes on first and the Class Ring above it toward the fingertip, keeping the Wedding Ring closest to the heart. One caution about this, however, many guys have the two rings soldered together so one doesn't rub away at the other. If you don't do this, you are looking at destroying one ring or both, especially if you get a 10K Class Ring and an 18K wedding ring.
As for me, its Wedding Ring on the left, moving the Class Ring to the right with the tradition of Academy Crest where I see it rather than my Class Crest. With this change, it does make it where I can read "West Point" and "1981" rather than someone I meet, but I kind of like it that way. It helps remind me where I come from, what I've experienced, and what I'm supposed to be. Also, like many West Pointers (contrary to popular belief) I do not try to shove my school up anyone's nose. In fact, most of my contemporaries used the codes "when I was at school", or "I went to school in New York" rather than be perceived as arrogant. I have since grown out of that with the understanding that I got a shot at some great opportunities, different from those I would have had at A&M, but still significant, and it's what we all make of those opportunities that matters.