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](https://en.wikipedia.org/wiki/Weyl_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 (in dimension $\geq 4$) 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.