Here, the structural symmetries of a hypergraph are represented through equivalence relations on the vertex set of the hypergraph. A matrix associated with the hypergraph may not reflect a specific structural symmetry. In the context of a given symmetry within a hypergraph, we investigate a collection of matrices that encapsulate information about the symmetry. Our investigation reveals that certain structural symmetries in a hypergraph manifest observable effects on the eigenvalues and eigenvectors of designated matrices associated with the hypergraph. We identify specific matrices where the invariance is a consequence of symmetries present in the hypergraph. These invariant subspaces elucidate analogous behaviours observed in certain clusters of vertices during random walks and other dynamical processes on the hypergraph.