Up to this point I have only been shooting right handed but am going to try left-handed due to some upper back problems from the muscle imbalance.  I took Rick Welches class and shoot right handed with an upright bow, double anchor, and dead release with a couple of seconds at anchor before release. When I do it right without mistakes I hit what I'm looking at. Because I am right eye dominant I'm considering shooting left handed with a canted bow to get my right eye over the arrow and shooting more with a follow through release. Sort of like I've seen Terry Green shoot on some of the fine videos he has shared on the site.  

Are there any of you that shoot from both sides that use a different style for each side? Any thoughts on this?

I don't think you're going to be able to get your right eye over the arrow no matter how much you cant your bow, because you're drawing to the left side of your face.

Just use the same good (though unorthodox) form Rick taught you when you shoot left handed, and if you shoot enough arrows, they will begin to work for you, although perhaps not as well as when you shoot with the arrow under your dominant eye.  

I enjoy shooting both right and left handed for the same reasons you mention.