  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.
