  The duality between partial actions (partial $H$-module algebras) and
co-actions (partial $H$-comodule algebras) of a Hopf algebra $H$ is fully
explored in this work. A connection between partial (co)actions and Hopf
algebroids is established under certain commutativity conditions. Moreover, we
continue this duality study, introducing also partial $H$-module coalgebras and
their associated $C$-rings, partial $H$-comodule coalgebras and their
associated cosmash coproducts, as well as the mutual interrelations between
these structures.
