  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) $Y\rightarrow X$. 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.
