Double divergence of a contravariant tensor ($T^{ab}{}_{;ab}=T^{ab}{}_{;ba}$)

For a Levi-Civita connection, a contravariant rank-two tensor satisfies $T^{ab}{}_{;ab}=T^{ab}{}_{;ba}$ without itself being symmetric. The Ricci identity gives their difference as $R_{pb}(T^{pb}-T^{bp})$, which vanishes because the Ricci tensor is symmetric. Zero torsion alone for a general affine connection does not ensure this symmetry.