A finite group with an integer representation has a multiplicative action on the ring of Laurent polynomials, which is induced by a nonlinear action on the complex torus. We study the structure of the associated orbit space as the image of the fundamental invariants. For the Weyl groups of types A, B, C and D, this image is a compact basic semi-algebraic set and we present the defining polynomial inequalities explicitly. We show how orbits correspond to solutions in the complex torus of symmetric polynomial systems and give a characterization of the orbit space as the positivity-locus of a symmetric real matrix polynomial. The resulting domain is the region of orthogonality for a family of generalized Chebyshev polynomials, which have connections to topics such as Fourier analysis and representations of Lie algebras.