We explore novel examples of RG flows preserving a non-invertible self-duality symmetry. Our main focus is on N=1 quadratic superpotential deformations of 4d N=4 super-Yang-Mills theory with gauge algebra su(N). A theory that can be obtained in this way is the so-called N=1* SYM where all adjoint chiral multiplets have a mass. Such IR theory exhibits a rich structure of vacua which we thoroughly examine. Our analysis elucidates the physics of spontaneous breaking of self-duality symmetry occurring in the degenerate gapped vacua. The construction can be generalized, taking as UV starting point a theory of class S, to demonstrate how non-invertible self-duality symmetries exist in a variety of N=1 SCFTs. We finally apply this understanding to prove that the conifold theory has a non-invertible self-duality symmetry.
