  We extend Turaev's definition of torsion invariants of 3-dimensional
manifolds equipped with non-singular vector fields, by allowing (suitable)
tangency circles to the boundary, and manifolds with non-zero Euler
characteristic. We show that these invariants apply in particular to (the
exterior of) Legendrian links in contact 3-manifolds. Our approach uses a
combinatorial encoding of vector fields, based on standard spines. In this
paper we extend this encoding from closed manifolds to manifolds with boundary.
