Doing a double Wick rotation in the world sheet theory of the light cone AdS5×S5 superstring results in an inequivalent, so-called mirror theory that plays a central role in the field of integrability in the AdS-CFT correspondence. We show that this mirror theory can be interpreted as the light cone theory of a free string on a different background. This background is related to dS5×H5 by a double T-duality, and has hidden supersymmetry. The geometry can also be extracted from an integrable deformation of the AdS5×S5 sigma model, and we prove the observed mirror duality of these deformed models at the bosonic level as a byproduct. While we focus on AdS5×S5, our results apply more generally.