  We derive a tensorial formula for a fourth-order conformally invariant
differential operator on conformal 4-manifolds. This operator is applied to
algebraic Weyl tensor densities of a certain conformal weight, and takes its
values in algebraic Weyl tensor densities of another weight. For oriented
manifolds, this operator reverses duality: For example in the Riemannian case,
it takes self-dual to anti-self-dual tensors and vice versa. We also examine
the place that this operator occupies in known results on the classification of
conformally invariant operators, and we examine some related operators.
