  Huang, Lepowsky and Zhang have developed a module theory for vertex operator
algebras that endows suitably chosen module categories with the structure of
braided monoidal categories. Included in the theory is a functor which assigns
to discretely strongly graded modules a contragredient module, obtained as a
gradewise dual. In this paper, we show that this gradewise dual endows the
module category with the structure of a ribbon Grothendieck-Verdier category.
This duality structure is more general than that of a rigid monoidal category;
in contrast to rigidity, it naturally accommodates the fact that a vertex
operator algebra and its gradewise dual need not be isomorphic as modules and
that the tensor product of modules over vertex operator algebras need not be
exact.
  We develop criteria which allow the detection of ribbon Grothendieck-Verdier
equivalences and use them to explore ribbon Grothendieck-Verdier structures in
the example of the rank $n$ Heisenberg vertex operator algebra or chiral free
boson on a not necessarily full rank even lattice with arbitrary choice of
conformal vector. We show that these categories are equivalent, as ribbon
Grothendieck-Verdier categories, to certain categories of graded vector spaces
and categories of modules over a certain Hopf algebra.
