  We describe the signatures of a circularly polarized gravitational-wave
background on the timing residuals obtained with pulsar-timing arrays. Most
generally, the circular polarization will depend on the gravitational-wave
direction, and we describe this angular dependence in terms of spherical
harmonics. While the amplitude of the monopole (the overall chirality of the
gravitational-wave background) cannot be detected, measures of the anisotropy
are theoretically conceivable. We provide expressions for the minimum-variance
estimators for the circular-polarization anisotropy. We evaluate the smallest
detectable signal as a function of the signal-to-noise ratio with which the
isotropic GW signal is detected and the number of pulsars (assumed to be
roughly uniformly spread throughout the sky) in the survey. We find that the
overall dipole of the circular polarization and a few higher overall
multipoles, are detectable in a survey with $\gtrsim100$ pulsars if their
amplitude is close to maximal and once the isotropic signal is established with
a signal-to-noise ratio $\gtrsim400$. Even if the anisotropy can be
established, though, there will be limited information on its direction.
Similar arguments apply to astrometric searches for gravitational waves.
