  We show that Fourier transforms on the Weyl algebras have a geometric
counterpart in the framework of toric varieties, namely they induce
isomorphisms between twisted rings of differential operators on regular toric
varieties, whose fans are related to each other by reflections of
one-dimensional cones. The simplest class of examples is provided by the toric
varieties related by such reflections to projective spaces. It includes the
blow-up at a point in affine space and resolution of singularities of varieties
appearing in the study of the minimal orbit of sl(n+1).
