  Within the scope of Bianchi type-$VI_0$ spacetime we study the role of spinor
field on the evolution of the Universe. It is found that the presence of
nontrivial non-diagonal components of energy-momentum tensor of the spinor
field plays vital role on the evolution of the Universe. As a result of their
mutual influence there occur two different scenarios. In one case the
invariants constructed from the bilinear forms of the spinor field become
trivial, thus giving rise to a massless and linear spinor field Lagrangian.
According to the second scenario massive and nonlinear terms do not vanish and
depending on the sign of coupling constants we have either an expanding mode of
expansion or the one that after obtaining some maximum value contracts and ends
in big crunch generating spacetime singularity. This result shows that the
spinor field is highly sensitive to the gravitational one.
