  We generalize Turaev's definition of torsion invariants of pairs $(M,\xi)$,
where $M$ is a 3-dimensional manifold and $\xi$ is an Euler structure on $M$ (a
non-singular vector field up to homotopy relative to the boundary of $M$ and
local modifications in the interior of $M$). Namely, we allow $M$ to have
arbitrary boundary and $\xi$ to have simple (convex and/or concave) tangency
circles to the boundary. We prove that Turaev's $H_1(M)$-equivariance formula
holds also in our generalized context. Our torsions apply in particular to (the
exterior of) Legendrian links (in particular, knots) in contact 3-manifolds,
and we prove that they can distinguish knots which are isotopic as framed knots
but not as Legendrian knots. Using the combinatorial encoding of vector fields
based on branched standard spines we show how to explicitly invert Turaev's
reconstruction map from combinatorial to smooth Euler structures, thus making
the computation of torsions a more effective one. As an example we work out a
specific computation.
