Consider a smooth manifold $M$ together with an affine connection (or covariant derivative) $\nabla$ on the tangent bundle $TM$.
Is it possible to have an affine connection, possibly with non-zero torsion, such that $\nabla_{X,Y}^2 = \nabla_{Y,X}^2$? I.e. the second covariant derivative is symmetric/commutative, while the covariant derivative itself is not (since it might have torsion)?
How are such connections or manifolds called? If anyone can refer me to an article or reference, that would be great.
PS. Note that the connection is not necessarily curvature-free in the case that $\nabla_{X,Y}^2 = \nabla_{Y,X}^2$, but instead all the curvature is due to torsion. That is, $R(X,Y)Z = \nabla_{T(X,Y)}Z$.