S-duality in $\mathcal{N}=1$ orientifold SCFTs

We present a general solution to the problem of determining all S-dual descriptions for a specific (but very rich) class of $\mathcal{N}=1$ SCFTs. These SCFTs are indexed by decorated toric diagrams, and can be engineered in string theory by probing orientifolds of isolated toric singularities with D3 branes. The S-dual phases are described by quiver gauge theories coupled to specific types of conformal matter which we describe explicitly. We illustrate our construction with many examples, including S-dualities in previously unknown SCFTs.