  Within the scope of Bianchi type-VI cosmological model we study the role of
spinor field in the evolution of the Universe. It is found that due to the
spinor affine connections the energy-momentum tensor of the spinor field
possesses non-diagonal components. The non-triviality of non-diagonal
components of the energy-momentum tensor imposes some severe restrictions
either on the spinor field or on the metric functions or on both of them. But
unlike in cases of Bianchi type-I or $VI_0$, in case of Bianchi type-VI model
it does not lead to the elimination of spinor field nonlinearity or mass term
in the spinor field Lagrangian. It is also found that depending on the sign of
self-coupling constant the model can give rise to late time acceleration or
generate oscillatory mode of evolution.
