The Riemann curvature tensor ${R^a}_{bcd}$ has a direct geometric interpretation in terms of parallel transport around infinitesimal loops.

Question: Is there a similarly direct geometric interpretation of the Weyl conformal tensor ${C^a}_{bcd}$?

Background: My understanding is that the Weyl conformal tensor is supposed to play a role in conformal geometry analogous to the role of the Riemann curvature tensor in (pseudo)Riemannian geometry. For instance, it is conformally invariant, and vanishes iff the manifold is conformally flat, just as the Riemann curvature tensor is a metric invariant and vanishes iff the manifold is flat. The two tensors also share many of the same symmetries. So it would be nice to have a more hands-on understanding of the Weyl tensor when studying conformal geometry.

Notes:

• I'd be especially happy with a geometric interpretation which is manifestly conformal in nature, referring not to the metric itself but only to conformally invariant quantities like angles.

• I'm also keen to understand any subtleties which depend on whether one is working in a Riemannian, Lorentzian, or more general pseudo-Riemannian context.