Geometric interpretation for conformally symmetric manifolds

If $(M, g)$ is a pseudo-Riemannian manifold, it’s well known that the condition $nabla R=0$ is equivalent to the existence of local geodesic symmetries about each point.

Another natural algebraic condition to impose on $(M,g)$ is for its Weyl tensor to be parallel: $nabla W = 0$.

It does not seem that there’s any analogue of Cartan’s theorem (quoted above) for this. This is hinted by the fact that while $W$ is a conformal invariant, the condition $nabla W = 0$ is not, as the Levi-Civita connection itself is not a conformal invariant.

How should one, in a very broad sense, geometrically understand $nabla W = 0$?