Let \({\mathcal S}(q,\Gamma)\) be the set of isometry classes of \(q-\)dimensional spherical space forms whose fundamental groups are isomorphic to a fixed group \(\Gamma\), and let \({\mathcal A}(q,\Gamma)\) be a certain group of transformations on the finite set \({\mathcal S}(q,\Gamma)\). It is shown that any two elements of the same \({\mathcal A}(q,\Gamma)-\)orbit are strongly isospectral. Since \({\mathcal A}(q,\Gamma)\) is generically nontrivial, this leads to a large set of new examples of isospectral non-isometric Riemannian manifolds. The results apply to low-dimensional spherical space forms and to homogeneous spherical space forms, and a part of them are carried over to Riemannian quotients of oriented real Grassmann manifolds.