  We generalize Turaev's definition of torsion invariants of pairs (M,x), where
M is a 3-dimensional manifold and x is an Euler structure on M (a non-singular
vector field up to homotopy relative to bM and local modifications in int(M).
Namely, we allow M to have arbitrary boundary and x 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) pseudo-Legendrian knots (i.e. knots
transversal to a given vector field), and hence to Legendrian knots in contact
3-manifolds. We show that torsion, as an absolute invariant, contains a lifting
to pseudo-Legendrian knots of the classical Alexander invariant. We also
precisely analyze the information carried by torsion as a relative invariant of
pseudo-Legendrian knots which are framed-isotopic. Using branched standard
spines to describe vector fields 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 calculation.
