  We introduce and study a canonical quadratic form, called the torsion
quadratic form, of the determinant line of a flat vector bundle over a closed
oriented odd-dimensional manifold. This quadratic form caries less information
than the refined analytic torsion, introduced in our previous work, but is
easier to construct and closer related to the combinatorial Farber-Turaev
torsion. In fact, the torsion quadratic form can be viewed as an analytic
analogue of the Poincare-Reidemeister scalar product, introduced by Farber and
Turaev. Moreover, it is also closely related to the complex analytic torsion
defined by Cappell and Miller and we establish the precise relationship between
the two. In addition, we show that up to an explicit factor, which depends on
the Euler structure, and a sign the Burghelea-Haller complex analytic torsion,
whenever it is defined, is equal to our quadratic form. We conjecture a formula
for the value of the torsion quadratic form at the Farber-Turaev torsion and
prove some weak version of this conjecture. As an application we establish a
relationship between the Cappell-Miller and the combinatorial torsions.
