= Solution
In the Klein fundamental square from part (b), the following straight lines project to the required closed geodesics:
$$
\gamma_1(t)=[t,0],
\qquad0\leq t\leq1;
$$
this is the one-sided horizontal core, and cutting along it leaves a Möbius strip.
Take
$$
\gamma_2(t)=[0,t-1/2],
\qquad0\leq t\leq1;
$$
this is a two-sided vertical geodesic, and cutting along it leaves a cylinder.
Finally take
$$
\gamma_3(t)=[2t,t],
\qquad0\leq t\leq1.
$$
Its endpoints differ by the deck translation $(2,1)$, so it is closed. The parameters $t=1/4$ and $t=3/4$ represent the same Klein-bottle point through an odd-$c$ glide reflection, and no other interior pair does. Thus its image has exactly one transverse self-intersection. This is the <flat Klein-bottle geodesic model>.
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