  Axial vectors, such as current or magnetization, are commonly used order
parameters in time-reversal symmetry breaking systems. These vectors also break
isotropy in three dimensional systems, lowering the spatial symmetry. We
demonstrate that it is possible to construct a fully isotropic and
inversion-symmetric three-dimensional medium where time-reversal symmetry is
systematically broken. We devise a cubic crystal with scalar time-reversal
symmetry breaking, implemented by hopping through chiral magnetic clusters
along the crystal bonds. The presence of only the spatial symmetries of the
crystal -- finite rotation and inversion symmetry -- is sufficient to protect a
topological phase. The realization of this phase in amorphous systems with
average continuous rotation symmetry yields a statistical topological insulator
phase. We demonstrate the topological nature of our model by constructing a
bulk integer topological invariant, which guarantees gapless surface spectrum
on any surface with several overlapping Dirac nodes, analogous to crystalline
mirror Chern insulators. We also show the expected transport properties of a
three-dimensional statistical topological insulator, which remains critical on
the surface for odd values of the invariant.
