The semiconductor Sr2FeOsO6, depending on temperature, adopts two types of spin structures that differ in the spin sequence of ferrimagnetic iron-osmium layers along the tetragonal c axis. Neutron powder diffraction experiments, 57Fe Mössbauer spectra, and density functional theory calculations suggest that this behavior arises because a lattice instability resulting in alternating iron-osmium distances fine-tunes the balance of competing exchange interactions. Thus, Sr2FeOsO6 is an example of a double perovskite, in which the electronic phases are controlled by the interplay of spin, orbital, and lattice degrees of freedom.