  Grothendieck-Verdier duality is a powerful and ubiquitous structure on
monoidal categories, which generalises the notion of rigidity. Hopf algebroids
are a generalisation of Hopf algebras, to a non-commutative base ring. Just as
the category of finite-dimensional modules over a Hopf algebra inherits
rigidity from the category of vector spaces, we show that the category of
finite-dimensional modules over a Hopf algebroid with bijective antipode
inherits a Grothendieck-Verdier structure from the category of bimodules over
its base algebra. We investigate the algebraic and categorical structure of
this duality.
