We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution $$\sigma $$ and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to define a generalization of the cohomological Donaldson–Thomas theory of quivers in which the quiver representations have orthogonal or symplectic structure groups. The associated invariants are called orientifold Donaldson–Thomas invariants. We prove the orientifold analogue of the integrality conjecture for $$\sigma $$ -symmetric quivers. We also formulate precise conjectures regarding the geometric meaning of orientifold Donaldson–Thomas invariants and the freeness of the CoHM of a $$\sigma $$ -symmetric quiver. We prove the freeness conjecture for disjoint union quivers, loop quivers and the affine Dynkin quiver of type $${\widetilde{A}}_1$$ . We also verify the geometric conjecture in a number of examples. Finally, we construct explicit Poincare–Birkhoff–Witt-type bases of the CoHM of finite type quivers.