In representation theory, a classical problem is to understand the decomposition of a given representation of a group G into irreducible summands. Multiplicities may appear and, when they do, the multiplicity spaces, on which G acts trivially, become representations of another algebraic structure, the centralizer of the representation. This additional information is often very useful, the first historical example being the Schur-Weyl duality. In this talk I will first discuss the known situation of triple tensor products of SU(2)-representations. In this case, the Racah algebra appears as the centralizer and this explains the appearance of the orthogonal Racah polynomials in this setting. Then I will present the algebra appearing for double tensor products of SU(3)-representations. Looking closely at this algebra, we discover an unexpected symmetry involving a Weyl group of type E6. I will explain how this has interesting consequences for concrete questions about tensor products of SU(3)-representations. This is a joint work with Nicolas Crampé and Luc Vinet.