  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 which allows the quiver representations to
have orthogonal and symplectic structure groups. The associated invariants are
called orientifold Donaldson-Thomas invariants. We prove the integrality
conjecture for orientifold Donaldson-Thomas invariants of $\sigma$-symmetric
quivers. We also formulate precise conjectures regarding the geometric meaning
of these 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 describe the CoHM
of finite type quivers by constructing explicit Poincar\'{e}-Birkhoff-Witt type
bases of these representations.
