  We consider unoriented strings in the supersymmetric SL(2,R)/U(1) coset,
which describes the two-dimensional Euclidean black hole, and its mirror dual
N=2 Liouville theory. We analyze the orientifolds of these theories from
several complementary points of view: the parity symmetries of the worldsheet
actions, descent from known AdS_3 parities, and the modular bootstrap method
(in some cases we can also check our results against known constraints coming
from the conformal bootstrap method). Our analysis extends previous work on
orientifolds in Liouville theory, the AdS_3 and SU(2) WZW models and minimal
models. Compared to these cases, we find that the orientifolds of the two
dimensional Euclidean black hole exhibit new intriguing features. Our results
are relevant for the study of orientifolds in the neighborhood of NS5-branes
and for the engineering of four-dimensional chiral gauge theories and gauge
theories with SO and Sp gauge groups with suitable configurations of D-branes
and orientifolds. As an illustration, we discuss an example related to a
configuration of D4-branes and O4-planes in the presence of two parallel
fivebranes.
