  We construct and classify categories of D-branes in orientifolds based on
Landau-Ginzburg models and their orbifolds. Consistency of the worldsheet
parity action on the matrix factorizations plays the key role. This provides
all the requisite data for an orientifold construction after embedding in
string theory. One of our main results is a computation of topological field
theory correlators on unoriented worldsheets, generalizing the formulas of Vafa
and Kapustin-Li for oriented worldsheets, as well as the extension of these
results to orbifolds. We also find a doubling of Knoerrer periodicity in the
orientifold context.
