  We discuss the consistency conditions of a novel orientifold projection of
type IIB string theory on C^2/Z_N singularities, in which one mods out by the
combined action of world-sheet parity and a geometric operation which exchanges
the two complex planes. The field theory on the world-volume of D5-brane probes
defines a family of six-dimensional RG fixed points, which had been previously
constructed using type IIA configurations of NS-branes and D6-branes in the
presence of O6-planes. Both constructions are related by a T-duality
transforming the set of NS-branes into the C^2/Z_N singularity. We also
construct additional models, where both the standard and the novel orientifold
projections are imposed. They have an interesting relation with orientifolds of
D_K singularities, and provide the T-duals of certain type IIA configurations
containing both O6- and O8-planes.
