  Though the Weyl tensor is a linear combination of the curvature tensor, Ricci
tensor and Ricci scalar, it does not have all and only the Lie symmetries of
these tensors since it is possible, in principle, that "asymmetries cancel".
Here we investigate if, when and how the symmetries can be different. It is
found that we can obtain a metric with a finite dimensional Lie algebra of Weyl
symmetries that properly contains the Lie algebra of curvature symmetries.
There is no example found for the converse requirement. It is speculated that
there may be a fundamental reason for this lack of "duality".
