  In this paper, we show that the Galois representations provided by
$\ell$-adic cohomology of proper smooth varieties, and more generally by
$\ell$-adic intersection cohomology of proper varieties, over any field, are
orthogonal or symplectic according to the degree. We deduce this from a
preservation result of orthogonal and symplectic pure perverse sheaves by
proper direct image. We show moreover that the subgroup of the Grothendieck
group generated by orthogonal pure perverse sheaves of even weights and
symplectic pure perverse sheaves of odd weights are preserved by Grothendieck's
six operations. Over a finite field, we deduce parity and symmetry results for
Jordan blocks appearing in the Frobenius action on intersection cohomology of
proper varieties, and virtual parity results for the Frobenius action on
ordinary cohomology of arbitrary varieties.
