  Cosmological birefringence, a rotation by an angle $\alpha$ of the
polarization of photons as they propagate over cosmological distances, is
constrained by the cosmic microwave background (CMB) to be
$|\alpha|\lesssim1^\circ$ ($1\sigma$) out to redshifts $z\simeq1100$ for a
rotation that is uniform across the sky. However, the rotation angle
$\alpha(\theta,\phi)$ may vary as a function of position $(\theta,\phi)$ on the
sky. Here I discuss how a position-dependent rotation can be sought in current
and future AGN data. An upper limit $\VEV{\alpha^2}^{1/2} \lesssim 3.7^\circ$
to the scatter in the position-angle--polarization offsets in a sample of only
N=9 AGN already constrains the rotation spherical-harmonic coefficients to
$(4\pi)^{-1/2} \alpha_{lm}\lesssim 3.7^\circ$ and constrains the power spectrum
for $\alpha$ in models where it is a stochastic field. Future constraints can
be improved with more sources and by analyzing well-mapped sources with a
tensor-harmonic decomposition of the polarization analogous to that used in CMB
polarization and weak gravitational lensing.
