  We have performed magnetic susceptibility, heat capacity, neutron powder
diffraction, and muon spin relaxation experiments to investigate the magnetic
ground states of the 5$d^2$ double perovskites Sr$_2$YReO$_6$ and
Sr$_2$InReO$_6$. We find that Sr$_2$YReO$_6$ is a spin glass, while
Sr$_2$InReO$_6$ hosts a non-magnetic singlet state. By making detailed
comparisons with other 5$d^2$ double perovskites, we argue that a delicate
interplay between spin-orbit coupling, non-cubic crystal fields, and exchange
interactions plays a key role in the great variation of magnetic ground states
observed for this family of materials.
