Besides its construction as a quotient of an abelian surface, a Kummer surface can be obtained as the quotient of a K3 surface by a Z/2Z-action. In this paper, we classify all such K3 surfaces. Our classification is expressed in terms of period lattices and extends Morrison's criterion of K3 surfaces with a Shioda–Inose structure. Moreover, we list all the K3 surfaces associated to a general Kummer surface and provide very geometrical examples of this phenomenon.