  A third order real tensor is mapped to a special f-diagonal tensor by going
through Discrete Fourier Transform (DFT), standard matrix SVD and inverse DFT.
We call such an f-diagonal tensor an s-diagonal tensor. 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. Then we present a set of sufficient and necessary conditions for
s-diagonal tensors. Such conditions involve a special complex number. In the
cases that the dimension of the third mode of the considered tensor is $2, 3$
and $4$, we present direct sufficient and necessary conditions which do not
involve such a complex number.
