  We consider an "orientifold" generalization of Khovanov-Lauda-Rouquier
algebras, depending on a quiver with an involution and a framing. Their
representation theory is related, via a Schur-Weyl duality type functor, to
Kac-Moody quantum symmetric pairs, and, via a categorification theorem, to
highest weight modules over an algebra introduced by Enomoto and Kashiwara. Our
first main result is a new shuffle realization of these highest weight modules
and a combinatorial construction of their PBW and canonical bases in terms of
Lyndon words. Our second main result is a classification of irreducible
representations of orientifold KLR algebras and a computation of their global
dimension in the case when the framing is trivial.
