  Motivated by Wigner's theorem, a canonical construction is described that
produces an Atiyah-Singer Dirac operator with both unitary and anti-unitary
symmetries. This Dirac operator includes the Dirac operator for KR-theory as a
special case, filling a long-standing gap in the literature. In order to make
the construction, orientifold Spin-c-structures are defined and classified
using semi-equivariant Dixmier-Douady theory, and analogues of several standard
theorems on the existence of Spin-c-structures are proved.
