In Riemannian geometry, when we think of the Ricci tensor merely as $Ric(\vec a, \vec b)$, then it's hard to describe it other than saying it's the contraction of the Riemann tensor. But when we realize that, due to its symmetry, it can be written as
$$Ric(\vec a, \vec b) = \frac12(Ric(\vec a + \vec b, \vec a + \vec b) - Ric(\vec a, \vec a) - Ric(\vec b, \vec b))$$
then we see it can be described entirely in terms of its action on single vectors, rather than pairs of vectors. And for such a component, we arrive at a really neat interpretation: it is the rate (acceleration?) of convergence of geodesics along that vector.
My question is: can we apply the same kind of logic to the Riemann tensor itself to get alternate, perhaps more intuitive, descriptions of it? (The difference, of course, is that we already have a useful interpretation before applying symmetry: we know that ${R^a}_{bcd}$ is "the $a$-component picked up by the $b$ vector upon parallel transport around the $cd$ loop")
Starting from that first description, we then have to lower the first index, and then, for example, we could focus on the symmetry of exchange of index pairs. In which case we're talking about components of the form $R_{abab}$. Is there a simple geometric description of such a component only in terms of the $ab$ loop (bivector?)? And could we arrive at other descriptions by applying the other symmetries? Or even by applying multiple symmetries at once?