<p>A closed contact manifold is called Besse when all its Reeb orbits are closed, and Zoll when they have the same minimal period. Motivated by the study of symmetric closed characteristics on symmetric compact convex hypersurfaces of Wang [22], we firstly introduce some variant concepts for symmetric convex contact spheres, i.e., a symmetric convex contact sphere is called <b>Symmetric-Besse</b> when all its Reeb orbits are symmetric, and <b>Symmetric-Zoll</b> when they have the same minimal period, then we provide a characterization for Symmetric-Besse and Symmetric-Zoll convex contact spheres in terms of the <i>S</i>¹-equivariant spectral invariants of symmetric Reeb orbits.</p>