  We study the uniqueness of minimal liftings of cut-generating functions
obtained from maximal lattice-free polyhedra. We prove a basic invariance
property of unique minimal liftings for general maximal lattice-free polyhedra.
This generalizes a previous result by Basu, Cornu\'ejols and K\"oppe~\cite{bcm}
for {\em simplicial} maximal lattice-free polytopes, thus completely settling
this fundamental question about lifting for maximal lattice-free polyhedra. We
further give a very general iterative construction to get maximal lattice-free
polyhedra with the unique-lifting property in arbitrary dimensions. This single
construction not only obtains all previously known polyhedra with the
unique-lifting property, but goes further and vastly expands the known list of
such polyhedra. Finally, we extend characterizations from~\cite{bcm} about
lifting with respect to maximal lattice-free simplices to more general
polytopes. These nontrivial generalizations rely on a number of results from
discrete geometry, including the Venkov-Alexandrov-McMullen theorem on
translative tilings and characterizations of zonotopes in terms of central
symmetry of their faces.
