This paper derives new identities for the Weyl tensor on a gradient Ricci soliton, particularly in dimension four. First, we prove a Bochner–Weitzenböck-type formula for the norm of the self-dual Weyl tensor and discuss its applications, including connections between geometry and topology. In the second part, we are concerned with the interaction of different components of Riemannian curvature and (gradient and Hessian of) the soliton potential function. The Weyl tensor arises naturally in these investigations. Applications here are rigidity results.