  We consider the precession of a Dirac particle spin in some anisotropic
Bianchi universes. This effect is present already in the Bianchi-I universe. We
discuss in some detail the geodesics and the spin precession for both the
Kasner and the Heckmann-Schucking solutions. In the Bianchi-IX universe the
spin precession acquires the chaotic character due to the stochasticity of the
oscillatory approach to the cosmological singularity. The related helicity flip
of fermions in the very early Universe may produce the sterile particles
contributing to dark matter.
