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We initiate the investigation of representation theory of non-orientablesurfaces. As a first step towards finding an additive categorification ofDupont and Palesi's quasi-cluster algebras associated marked non-orientablesurfaces, we study a certain modification on the objects of the clustercategory associated to the orientable double covers in the unpunctured case.More precisely, we consider symmetric representation theory studied byDerksen-Weyman and Boos-Cerulli Irelli, and lift it to the cluster category.This gives a way to consider `indecomposable orbits of objects' under acontravariant duality functor. Hence, we can assign curves on a non-orientablesurface $(\mathbb{S}, \mathbb{M})$ to indecomposable symmetric objects.Moreover, we define a new notion of symmetric extension, and show that the arcsand quasi-arcs on $(\mathbb{S}, \mathbb{M})$ correspond to the indecomposablesymmetric objects without symmetric self-extension. Consequently, we show thatquasi-triangulations of $(\mathbb{S}, \mathbb{M})$ correspond to a symmetricanalogue of cluster tilting objects.

Marked non-orientable surfaces and cluster categories via symmetric representations