  Oxygen vacancies in the perovskite insulator SrTiO$_3$ free electrons that
couple with other physical degrees of freedom such as lattice, orbital, and
spin. This leads to the emergence of exotic quantum states such as
superconductivity and unusual ferromagnetism. We perform density-functional
theory and dynamical mean-field theory calculations and demonstrate that the
orientation and ordering of the TiO$_5$ pentahedra plays a crucial role.
Specifically, for vacancy-rich SrTiO$_{3-\delta}$ ($\delta\sim$0.5), we find a
chiral ordering of the TiO$_5$ pentahedra in a sixfold superlattice. This
chiral structure is accompanied by a chiral magnetic state with a net moment in
the (111) direction at room temperature, which can explain several experimental
observations.
