Reduce ATM to the language of TM encodings concatenated by a string where the TM accepts both the string and its reverse
Hint
Forget the similarity between $LM$ and $A_{\mathrm{TM}}$, and for an input $\langle M,x\rangle$ of the reduction, try to build a new TM $N$ such that $M$ accepts $x$ if and only if $N$ accepts $0$, i.e. $\langle N,0\rangle\in LM$.
Complete answer

 Given $\langle M,x\rangle$, build a TM $N_{M,x}$ that on any input always simulates $M$ on $x$. Therefore if $M$ accepts $x$, $N_{M,x}$ accepts everything, otherwise $N_{M,x}$ rejects anything. Now we can see $M$ accepts $x$ if and only if $N_{M,x}$ accepts $0$, i.e. $\langle N_{M,x},0\rangle\in LM$. All above completes the reduction.