Are there analogues to orthogonal transformations in non-orientable surfaces?
Bear with me, as this is really only tangential¹ to my area of expertise, but:

Maybe the question is not how to extend wavelets to work on non-orientable surfaces, but to interpret said surface as something where wavelets natively work well.

Enter differential geometry (differential topology? I don't know.):

As you've notice: if you only consider a part of your moebius strip, let's say, everything but the original "yellow line", then it looks perfectly much like structure that you can work with: a (discrete) two-dimensional vector space. I presume you have some explicitly vector-space notation of what your matrix represents!

In fact, the idea here would be to treat your surface as a vector bundle: 
You take a circle (which is, if you think about it, a one-dimensional vector space itself, with some kind of "modulo" characteristics). To every point on that circle, you assign yet another vector space: a one dimensional line, tangential to that circle. 

Now, take a part of your circle (an arc), and find the product space of that with your vector bundle: That's something that should look topologically like your original vector space.

This way of looking at your moebius strip should allow you to use your transforms like you used to use it on the original matrix.


¹ hrhr, a pun.