Before revision:  We show that an f-diagonal tensor is an s-diagonal tensor if and only if ST-TVD factorization diagonalizes it to itself . Four meaningful necessary conditions for s-diagonal tensors are presented.
After revision:  We show that an f-diagonal tensor is an s-diagonal tensor if and only if it is mapped to itself in the above process. The third order tensor space is partitioned to orthogonal equivalence classes. Each orthogonal equivalence class has a unique s-diagonal tensor. Two s-diagonal tensors are equal if they are orthogonally equivalent. Third order tensors in an orthogonal equivalence class have the same tensor tubal rank and T-singular values . Four meaningful necessary conditions for s-diagonal tensors are presented.