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In the Klein fundamental square from part (b), the following straight lines project to the required closed geodesics:this is the one-sided horizontal core, and cutting along it leaves a Möbius strip.

Takethis is a two-sided vertical geodesic, and cutting along it leaves a cylinder.

Finally takeIts 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.