Mmmmh the title may not be that appropriate relatively to the message I wanted to convey. Here, I am not addressing mirror worlds in the sense of a multiverse thing. I am considering a single universe (i.e. ours) that contains two secluded sectors of particles, that I have abusively called worlds.
One of these is made of matter and its interactions, the second one is made of dark particles and their own interactions. Thanks to very weak mediations, a given sector is allowed to speak with the other one (which provides a way to the visible side to probe the dark side, and vice versa).
So after this intermezzo, to go back to your question:
Could a "mirror world" also just be our own universe rotated slightly in an alternative spacial dimension
Yes, it could somehow, except than instead of talking about 'dimensions', we could speak about 'ways to interact'.
[..] with our only interaction being gravitational?
Gravity is always there of course, but too weak to play any role at the particle level. The funny thing is that the Higgs boson and its dark counterpart are by definition making a connection. Similarly, dark electromagnetism and electromagnetism are connected automatically. Therefore, if one takes a mirror of the Standard Model, there are two connections beyond gravity by construction. Those connections are way stronger than gravity (even if they are weak, gravity is way weaker), and they are those that can be constrained experimentally.
I hope this answers at least a little your question. Don't hesitate to come back to me if needed :)
I indeed do not talk bout extra dimensions (which is en entirely other topic). In this post, I consider our good old four-dimensional space-time. There are just two classes of particles in there, and interactions between classes are extremely feeble.
The signal to observe would here be only this feeble interaction happening in a detector.
Does it clarify?