The Weyl group of a crystallographic root system has a multiplicative action on the ring of multivariate trigonometric polynomials. We study the problem of minimizing an invariant trigonometric polynomial. This problem can be written as a polynomial optimization problem on a compact basic semi-algebraic set. Lasserre's moment-SOS hierarchy is formulated in the basis of generalized Chebyshev polynomials, leading to a different notion of the polynomial degree and smaller matrices in the arising semi-definite program. Bilevel optimization techniques are applied to solve max-min problems. Optimal values of trigonometric polynomials appear in spectral bounds for chromatic numbers and independence numbers of geometric distance graphs. We study the quality of these bounds for polytope norms and prove sharpness in several cases.