  We present a general framework to exploit the symmetries present in the
Navascu{\'e}s-Pironio-Ac{\'i}n semidefinite relaxations that approximate
invariant noncommutative polynomial optimization problems. We put equal
emphasis on the moment and sum-of-squares dual approaches, and provide a
pedagogical and formal introduction to the Navascu{\'e}s-Pironio-Ac{\'i}n
technique before working out the impact of symmetries present in the problem.
Using our formalism, we compute analytical sum-of-square certificates for
various Bell inequalities, and prove a long-standing conjecture about the exact
maximal quantum violation of the CGLMP inequalities for dimension 3 and 4. We
also apply our technique to the Sliwa inequalities in the Bell scenario with
three parties with binary measurements settings/outcomes. Symmetry reduction is
key to scale the applications of the NPA relaxation, and our formalism
encompasses and generalizes the approaches found in the literature.
