Spectral hypergraph theory studies the structural properties of a hypergraph that can be inferred from the eigenvalues and the eigenvectors of either matrices or tensors associated with it. In this paper we study the spectral indistinguishability in the hypergraph setting. We present a general switching method to construct uniform $E$-cospectral hypergraphs (hypergraphs with the same $E$-spectrum), and discuss some of its multiple applications. Our method not only provides a framework to unify the existing methods for obtaining $E$-cospectral hypergraphs via switching, but also generalizes most of the existing switching tools, yielding multiple new constructions. Finally, we compare common methods of computing $E$-characteristic polynomials, and in particular show that one standard method, while useful for generic tensors, is uninformative for almost all hypergraphs.