It just so happens I modeled a Mobius strip a little while back, and I'm sure the technique I used can address your case as well.
Model a plane, or some variation of a plane. Best to keep it simple at first.
Place an Empty at the end of your object. Actually, it should be a little further from the end of your object, but we'll adjust it later.
Select the Empty, then the mesh object.
Edit Mode, with the end vertices selected, Hook those verts to the Empty Ctrl + H (Hook to Selected Object).
Add an Array Modifier to your plane/mesh object. Choose
Object Offset and point it to the Empty. Set the Count to 3 or 4 iterations so you can see the effect.
Object Mode rotate that Empty and you'll see that mesh twist like a DNA double-helix.
This will probably need some fine-tuning, so in
Edit Mode with those same end verts selected you can un-hook the verts (also Ctrl + H]). Then move the Empty and re-hook the verts. You can play with it until the positioning is right. It's hard to describe this part, but if you try it you'll get it.
Finally, you should have a strip that you can twist and stretch using the Empty.
Once you've twisted the strip to the degree suitable for your design, save your file, then apply the modifiers.
Place your 3D cursor where the center of your ring will be and align your view accordingly, then use the classic warp tool -
Warp to wrap your mesh into a ring shape. Remove Doubles.
Another way to to this would be to use the Curve Modifier, but it won't give you an evenly incremented twist like this technique does.
For anyone creating a (Mobius strip like) twist where the normals oppose each other where they meet, there is the additional challenge of how to merge these, which I will briefly address. Here, the vertices at the edge where the normals are opposite have been ripped with pressing V.
Halving the Solidify Modifier's Thickness value, then duplicating the object and making the value negative on the duplicate is one way to fix this. Then you can apply the Modifiers, join the two objects, and merge the vertices at the seam. After that, additional clean-up of of the mesh, such as the deletion of internal faces may also be necessary.
Once the two objects have been joined, and vertices welded, here is the final result:
Two .blend files for reference: