EQtm is not mapping reducible to its complement This is a problem from Sipser's book (marked with an asterisk). 

$EQ_{TM} = \{(\langle M \rangle, \langle N \rangle)$ where $M$ and $N$ are Turing machines and $L(M) = L(N)\}$

We know that neither $EQ_{TM}$ nor $\overline{EQ_{TM}}$ are recognizable so unsure how to go about proving there can't be a mapping reduction from one to the other.

Any hints?
Your language is $\Pi_2$-complete: it can be written as a $\Pi_2$ formula, and TOT can be reduced to it. The complement is therefore $\Sigma_2$-complete. No computable reductions can exist between $\Pi_2$-complete and $\Sigma_2$-complete sets (why?).