We study duality for monoids in an infinite-dimensional setting that are invariant under the action of the infinite symmetric group Sym. Our main result is an equivariant Minkowski--Weyl theorem for monoids. More precisely, we analyze the evolution of dual monoids along stabilizing Sym-invariant chains and describe the eventual behavior of their equivariant Hilbert bases. In addition, we develop a systematic study of structural properties of dual symmetric monoids, including a characterization of the duals of positive and non-positive monoids.