If two theories A and B are mutually interpretable, in the sense of there existing a translation procedure from A to B and from B to A, does it follow that whatever metatheoretic results (e.g., categoricity) hold for one theory hold for the other?
If it is not true generally, does it hold for some subset of metatheoretic results? For instance, I know that mutual interpretability implies equiconsistency (see, e.g., Colin McLarty's second comment here). Intuitively, it seems like categoricity should carry over if it is right to think of failure of categoricity as resulting from a lack of expressive power (like how first order theories often lack categoricity due to an inability to discriminate between different sized infinite models and thus admitting of Skolemized models). Is this much right, at least?