How to prove the set of Turing machines that accept a string and its mirror is undecidable? I'm trying to prove the undecidability of the following language. 

$$L=\{\langle M \rangle\mid M\text{ is a Turing machine and there is a string }w\\\text{ s.t. }M\text{ accepts }w\text{ and }M\text{ rejects }w'\},$$ 
where $w'$ is the mirrored version of $w$.

I know that my first steps should be to find a reduction from $A_{\text{TM}}$, which is undecidable but the rejection part of the mirrored string is proving troublesome.
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.