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We study orientifold projections of families of four-dimensional$\mathcal{N}=1$ toric quiver gauge theories. We restrict to quivers that havethe unusual property of being associated with multiple periodic planar diagramswhich give rise, in general, to inequivalent models. A suitable orientifoldprojection relates a subfamily of the latter by conformal duality. That is,there exist exactly marginal deformations that connect the projected models.The deformations take the form of a sign flip in some of the superpotentialinteractions, similarly to the $\beta$-deformation of $\mathcal{N}=4$ SYM. Ourconstruction generalizes previous results on the orientifold projections of thePdP$_{3b}$ and PdP$_{3c}$ singularities.

Multi-planarizable quivers, orientifolds, and conformal dualities