Accurate and generalizable Reynolds-averaged Navier-Stokes (RANS) models for turbulent flows rely on effective closures, but currently available closures are notoriously unreliable. Kassinos et al. (J. Fluid Mechanics, 428, pp. 213-248, 2001) hypothesized that this unreliability of RANS models was due to an insufficient description of the statistical state of the turbulence and proposed a set of structure tensors as a candidate for a sufficiently rich description. To test this hypothesis for the rapid pressure-strain term, we introduce tensor-based, symmetry aware closures in terms of the structure tensors using equivariant neural networks (ENNs), and present an algorithm for enforcing algebraic contraction relations among tensor components. Using data from rapid distortion theory, experiments show that such ENNs can effectively learn relationships involving high-order tensors. The resulting ENN structure tensor models are orders of magnitude more accurate than existing models for the rapid pressure-strain correlation, effectively validating the Kassinos et al. hypothesis for this term. Results show that ENNs provide a physically consistent alternative to classical tensor basis models, enabling end-to-end learning of unclosed terms in RANS and other tensor modeling domains, and rapid exploration of model dependencies.