  We study orientifold projections of families of four-dimensional
$\mathcal{N}=1$ toric quiver gauge theories. We restrict to quivers that have
the unusual property of being associated with multiple periodic planar diagrams
which give rise, in general, to inequivalent models. A suitable orientifold
projection 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 superpotential
interactions, similarly to the $\beta$-deformation of $\mathcal{N}=4$ SYM. Our
construction generalizes previous results on the orientifold projections of the
PdP$_{3b}$ and PdP$_{3c}$ singularities.
