  We prove the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for a
measure supported on abelian differentials which come from non-orientable
quadratic differentials through a standard orientating, double cover
construction. The proof uses Forni's criterion for non-uniform hyperbolicity of
the cocycle for SL(2,R)-invariant measures. We apply these results to the study
of deviations in homology of typical leaves of the vertical and horizontal
(non-orientable) foliations and deviations of ergodic averages.
