How to prove the set of Turing machines that accept a string and its mirror is undecidable?
It's the standard reduction from the Halting problem: given a machine $M$ and an input $w$ to it, construct a new machine $M$ that accepts your favourite string and rejects its reverse if $M(w)$ halts, and accepts nothing if $M(w)$ does not halt.

This technique applies to any question of the form "Prove that it's undecidable whether a Turing machine accepts a string or language of some specific kind."

Or just use Rice's theorem, if you've been taught that.