Real algebraic geometry studies sets defined by a finite system of real polynomial equalities and inequalities. A central topic in this area is the study of the cone of nonnegative polynomials. Verifying that a given polynomial is nonnegative is an NP-hard problem. However, it turns out to be algorithmically much more feasible to verify if a given polynomial admits a representation into a sum of squares of polynomials, and such a decomposition provides a certificate for nonnegativity. Therefore, understanding the sets of sums of squares and nonnegative polynomials provides applications to various fields such as polynomial optimization and graph theory. In this thesis, tropicalization and the combinatorics of reflection groups are exploited to examine the cones of invariant sums of squares forms and nonnegative forms, and to study invariant systems of equations. In these contexts the thesis provides new insights in the description of sums of squares invariant under infinite series of essential reflection groups, the asymptotic behavior of the cones of symmetric and even symmetric sums of squares and nonnegative forms, and the ideals corresponding to the irreducible representations of the hyperoctahedral group.