Algebraic models of symmetric Nash sets

The aim of this paper is to prove the existence of algebraic models for Nash sets having suitable symmetries. Given a Nash set $M \subset \R^n$, we say that $M$ is specular if it is symmetric with respect to an affine subspace $L$ of $\R^n$ and $M \cap L=\emptyset$. If $M$ is symmetric with respect to a point of $\R^n$, we call $M$ centrally symmetric. We prove that every specular compact Nash set is Nash isomorphic to a specular real algebraic set and every specular noncompact Nash set is semialgebraically homeomorphic to a specular real algebraic set. The same is true replacing ``specular'' with ``centrally symmetric'', provided the Nash set we consider is equal to the union of connected components of a real algebraic set. Less accurate results hold when such a union is symmetric with respect to a plane of positive dimension and it intersects that plane. The algebraic models for symmetric Nash sets $M$ we construct are symmetric. If the local semialgebraic dimension of $M$ is constant and positive, then we are able to prove that the set of birationally nonisomorphic symmetric algebraic models for $M$ has the power of continuum.