  We explore novel examples of RG flows preserving a non-invertible
self-duality symmetry. Our main focus is on $\mathcal{N}=1$ quadratic
superpotential deformations of 4d $\mathcal{N}=4$ super-Yang-Mills theory with
gauge algebra $\mathfrak{su}(N)$. A theory that can be obtained in this way is
the so-called $\mathcal{N}=1^*$ 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
$\mathcal{S}$, to demonstrate how non-invertible self-duality symmetries exist
in a variety of $\mathcal{N}=1$ SCFTs. We finally apply this understanding to
prove that the conifold theory has a non-invertible self-duality symmetry.
