  We study various orientifold projections of 4d $\mathcal{N}=1$ toric gauge
theories, associated with CY singularities known as $L^{a,b,a}/\mathbb{Z}_2$,
with $a+b$ even. We obtain superconformal chiral theories that have the same
central charge, anomalies and superconformal index, whereas they were different
before the orientifold. Some of these projections are implemented by a novel
type of orientifold without fixed loci, known as glide orientifold. We claim
that these theories flow to the same conformal manifold, and they are connected
by quadratic exactly marginal deformations. The latter can be written in terms
of conjugate pairs of bifundamental fields of $R$-charge one, generalizing
previous results for unoriented non-chiral theories.
