  We prove a twisted Poincar\'e duality for (full) Hopf algebroids with
bijective antipode. As an application, we recover the Hochschild twisted
Poincar\'e duality of Van Den Bergh
  [VDB]. We also get a Poisson twisted Poincar\'e duality, which was already
stated for oriented Poisson manifolds in [CLYZ].
