= Solution
Apply the first extraction move to the two outer occurrences of $x$, then cyclically rotate the word to expose the two occurrences of $y^{-1}$:
$$
\begin{aligned}
xyzzyx
&\sim xx\,y^{-1}z^{-1}z^{-1}y^{-1}\\
&\sim y^{-1}z^{-1}z^{-1}y^{-1}xx\\
&\sim y^{-1}y^{-1}zzxx\\
&\sim xx\,y^{-1}y^{-1}zz.
\end{aligned}
$$
In the third line the repeated letter for the move is $y^{-1}$, and the intervening string is $z^{-1}z^{-1}$, whose inverse is $zz$. Rename $y^{-1}$ as a new edge label. The final word has the standard form $aabbcc$.
Each same-direction repeated pair represents a projective-plane summand, so the <classification theorem for surfaces> identifies this word with the <connected sum> of three real projective planes:
$$
\boxed{S\cong\mathbb{RP}^2\mathbin{\#}\mathbb{RP}^2\mathbin{\#}\mathbb{RP}^2.}
$$
Thus the surface has <crosscap number> three. As a check, the original identifications make all six polygon vertices equivalent, leaving one <vertex>, three edges and one face. Its <Euler characteristic> is $1-3+1=-1=2-3$, consistent with that crosscap count.
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