  In this paper we define a Poincar\'e-Reidemeister scalar product on the
determinant line of the cohomology of any flat vector bundle over a closed
orientable odd-dimensional manifold. It is a combinatorial "torsion-type"
invariant which refines the PR-metric, introduced earlier by the first author,
and contains an additional sign or phase information. We compute the PR-scalar
product in terms of the torsions of Euler structures, introduced earlier by the
second author. We show that the sign of our PR-scalar product is determined by
the Stiefel-Whitney classes and the semi-characteristic of the manifold. As an
application, we compute the Ray-Singer analytic torsion via the torsions of
Euler structures. Another application: a computation of the twisted
semi-characteristic in terms of the Stiefel-Whitney classes.
