TITLE: Interpretation of Riemann tensor antisymmetry
QUESTION [5 upvotes]: We know that the Riemann tensor is antisymmetric with respect to the first two vectors (the vectors that we parallel transport the third vector around the parallelogram made by their integral curves).
But what is the geometric interpretation?
Why do we have to expect that by changing the order of parallel transportation we will have the same vector with opposite orientation?

REPLY [7 votes]: More interpretations:
On (pseudo-) Riemannian manifolds:
the numerator of sectional curvature $-\langle R(X,Y)X,Y\rangle$ is a symmetric bilinear form
on the space of skew-symmetric bivectors. Skew symmetric bivectors describe measured 2-planes in the tangent space. Curvature in the form of  $\langle R(X,Y)Z,W\rangle$ can be recomputed by polarization from this.
Another, more general point of view: On the (orthonormal) frame bundle, curvature is a 2-form with values in the Lie algebra of the structure group: $\Omega=d\omega+\omega\wedge\omega$ for matrix valued forms. This ties in well with the fact that curvature is the obstruction against integrability of the horizontal subbundle of $TTM$.