  This work is a spin-off of an on-going programme which aims at revisiting the
original studies of Lie and Cartan on pseudogroups and geometric structures
from a modern perspective. Within the framework of Lie groupoids equipped with
a special multiplicative form - called Pfaffian groupoids - we focus on
principal bibundles and Morita equivalences. In particular, we discuss in
details the notion of Pfaffian Morita equivalence, its relation to the gauge
construction in the Pfaffian setting, and its interactions with principal
actions. We briefly present some examples and applications to transitive
pseudogroups of symmetries, which we explored in great detail in
arXiv:2211.16639.
