  We associate a family of cluster X-varieties to the dual Poisson-Lie group G*
of a complex semi-simple Lie group G of adjoint type given with the standard
Poisson structure. This family is described by the W-permutohedron associated
to the Lie algebra g of G: vertices being labeled by cluster X-varieties and
edges by new Poisson birational isomorphisms, on appropriate seed X-tori,
called saltation. The underlying combinatorics is based on a factorization of
the Fomin-Zelevinsky twist maps into mutations and other new Poisson birational
isomorphisms on seed X-tori called tropicalmutations, associated to an
enrichment of the combinatorics on double words of the Weyl group W of G.
