  Given a Lie groupoid, we can form its orbit space, which carries a natural
diffeology. More generally, we have a quotient functor from the
Hilsum-Skandalis category of Lie groupoids to the category of diffeological
spaces. We introduce the notion of a lift-complete Lie groupoid, and show that
the quotient functor restricts to an equivalence of the categories: of
lift-complete Lie groupoids with isomorphism classes of surjective submersive
bibundles as arrows, and of quasi-\'{e}tale diffeological spaces with
surjective local subductions as arrows. In particular, the Morita equivalence
class of a lift-complete Lie groupoid, alternatively a lift-complete
differentiable stack, is determined by its diffeological orbit space. Examples
of lift-complete Lie groupoids include quasifold groupoids and \'{e}tale
holonomy groupoids of Riemannian foliations.
