  We study the geometry of orientifolds in the SU(2) WZW model. They correspond
to the two inequivalent, orientation-reversing involutions of $S^3$, whose
fixed-point sets are: the north and south poles (O0), or the equator two-sphere
(O2). We show how the geometric action of these involutions leads unambiguously
to the previously obtained algebraic results for the Klein bottle and Moebius
amplitudes. We give a semiclassical derivation of the selection rules and signs
in the crosscap couplings, paying particular attention to discrete B-fluxes.
  A novel observation, which does not follow from consistency of the one-loop
vacuum diagrams, is that in the case of the O0 orientifolds only integer- or
only half-integer-spin Cardy states may coexist.
