= Solution
Label the endpoints around the original polygon by $v_0,v_1,v_2,v_3$ so that the successive <boundary> arcs are
$$
v_0\xrightarrow{x}v_1\xrightarrow{A}v_2\xrightarrow{x}v_3\xrightarrow{B}v_0.
$$
The two occurrences of $x$ have the same boundary-traversal direction, as prescribed by the word. We use cutting along a new diagonal followed by gluing the old $x$ sides. Cutting introduces two copies of the diagonal with opposite orientations, so rejoining them later exactly recovers the original quotient.
\b[First move.] Draw a diagonal $d$ from $v_0$ to $v_2$. The two pieces have <boundary> words $xAd^{-1}$ and $dxB$. Glue them along their $x$ sides, turning one piece over in the planar drawing to match the arrows. Reading the outside <boundary> of the joined disk gives
$$
Ad^{-1}d^{-1}B^{-1}.
$$
Reverse the orientation of this entire disk and change the starting point of its <boundary> word. The same side-paired disk then has word $ddA^{-1}B$. Renaming the new paired edge $d$ as $x$ yields
$$
\boxed{xAxB\sim xxA^{-1}B.}
$$
\b[Second move.] Instead cut along $d$ from $v_1$ to $v_3$. The pieces now have words $xdB$ and $Axd^{-1}$. Gluing along the old $x$ sides gives outside word $dBA^{-1}d$. A cyclic change of starting point gives $A^{-1}ddB$, and renaming $d$ yields
$$
\boxed{xAxB\sim A^{-1}xxB.}
$$
All side identifications inside the strings $A$ and $B$ are carried along unchanged; reversing a piece reverses the order of its string and inverts each letter. The construction therefore gives homeomorphic quotients, not merely an equality of Euler characteristics. This is the <crosscap extraction move for a polygonal word>.
\Image[/past-exam-of-the-topology-course-of-the-university-of-oxford/2014-crosscap-slide.png]
{title=Cutting a diagonal and gluing the old paired sides extracts a crosscap}
{description=Each row uses one of the two diagonals. The right-hand joined disks keep the old x seam inside and expose a new paired diagonal on the boundary. Reversing or cyclically reading that boundary gives the two required word forms.}
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