An orientifold is a background for string sigma-model that combines aspects of -orbifolds with orientation reversal on the worldsheet (therefore the name): it consists of a bundle gerbe on a space with a -action that satisfies a peculiar twisted equvariance condition with respect to this action.
Such orientifold gerbes with connection are the right structure for the definition of surface holonomy of unoriented surfaces. Therefore they serve for defining the gauge part of the action functional for unoriented strings.
One formalization of orientifold backgrounds appeared in
Urs Schreiber, Christoph Schweigert, Konrad Waldorf, Unoriented WZW models and Holonomy of Bundle Gerbes (arXiv)
Krzysztof Gawedzki, Rafal R. Suszek, Konrad Waldorf, Bundle Gerbes for Orientifold Sigma Models (arXiv)
A formalization that also takes the spinorial degrees of freedom into account is in
A formulation in differential nonabelian cohomology is indicated in