Let $G$ be a non-abelian finite group of rank 2 defined from generators $A$ and $B$.
For any element of $G$ defined by a word $W$, let the dual of that element be defined by a word $\overline{W}$, formed by replacing each of the letters $A$ and $B$ with the other, e.g. $AABBAB$ becomes $BBAABA$.
If every valid equation relating elements remains true when the generating elements are swapped, that is $W=V\iff \overline{W}=\overline{V} $ (e.g. $ABB=BAAB \implies BAA=ABBA$, then what can be concluded about $G$?
Do such groups exist? and if so what is the order of their irreducible faithful matrix representation.