The double of the doubles of Klein surfaces

A Klein surface is a surface with a dianalytic structure. A double of a Klein surface X is a Klein surface Y such that there is a degree two morphism (of Klein surfaces) Y -> X. There are many doubles of a given Klein surface and among them the so-called natural doubles which are: the complex double, the Schottky double and the orienting double. We prove that if X is a non-orientable Klein surface with non-empty boundary, the three natural doubles, although distinct Klein surfaces, share a common double: "the double of doubles" denoted by DX. We describe how to use the double of doubles in the study of both moduli spaces and automorphisms of Klein surfaces. Furthermore, we show that the morphism from DX to X is not given by the action of an isometry group on classical surfaces.