= Crosscap extraction move for a polygonal word
For a <polygonal schema> with all its sides paired and two same-direction occurrences of an edge label $x$, the words $xAxB$, $xxA^{-1}B$, and $A^{-1}xxB$ describe <homeomorphic> <quotient spaces>. Here inversion of a string reverses its order and inverts each letter. Cut from the start of the first $x$ side to the start of the second, using a diagonal $d$. The pieces have boundary words $xAd^{-1}$ and $dxB$. Gluing the old $x$ sides gives outside word $Ad^{-1}d^{-1}B^{-1}$; reversing the whole boundary and choosing a new starting point gives $ddA^{-1}B$. Alternatively cut from the end of the first $x$ side to the end of the second. The pieces are $xdB$ and $Axd^{-1}$, and gluing their old $x$ sides gives $dBA^{-1}d$, cyclically $A^{-1}ddB$. Renaming the new paired side gives both asserted forms. These are actual cutting and gluing <homeomorphisms>, so they preserve all the other side identifications as well as the <Euler characteristic>.
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