A generalization of fixed-lattice cluster expansions (CE) and atomic cluster expansions (ACE) is presented that expresses both rotation and permutation symmetries. By performing Schur–Weyl decomposition in Young subgroup-stabilized carriers, followed by joint coupling of rotation and permutation representations, arbitrary S_N× SO(3) tensor product carriers are generated. This allows us to resolve complete, orthonormal joint rotation and permutation-adapted bases for arbitrary tensor product ranks and arbitrary angular and radial tensor product content. We show that this very general Young–E(3) (YE3T) tensor product basis contains the ACE basis as a subset, corresponding to the special case where permutation symmetry character is restricted to the fully symmetric carrier. Rather than constructing an overcomplete rotation and permutation-invariant basis and reducing it a posteriori, Barthelemy et al. recently demonstrated scaling benefits by not constructing an overcomplete basis. YE3T directly produces a complete orthonormal basis without an overcomplete step and rigorously extends to permutation characters beyond permutation-symmetric features. The YE3T decomposition yields new joint irreducible rotation- and permutation-equivariant basis sets that surpass existing rotation-adapted expansions in both speed and accuracy, defining a new Pareto front for machine-learned interatomic potentials. We show that the tunability of the permutation symmetry character makes the basis useful for both atomistic and electronic-structure simulations.