Given a simple algebraic group $G$ and a parabolic subgroup $P\subset G$, with maximal torus $T$, we consider the closure $X$ of a generic $T$-orbit (in the sense of Dabrowski's work) in $G/P$, and determine when $X$ is a Gorenstein-Fano variety. We deduce of this classification a list of some pairs of dual reflexive polytopes.