  We present a search for anisotropic cosmic birefringence in 500 deg$^2$ of
southern sky observed at 150 GHz with the SPTpol camera on the South Pole
Telescope. We reconstruct a map of cosmic polarization rotation anisotropies
using higher-order correlations between the observed cosmic microwave
background (CMB) $E$ and $B$ fields. We then measure the angular power spectrum
of this map, which is found to be consistent with zero. The non-detection is
translated into an upper limit on the amplitude of the scale-invariant cosmic
rotation power spectrum, $L(L+1)C_L^{\alpha\alpha}/2\pi < 0.10 \times 10^{-4}$
rad$^2$ (0.033 deg$^2$, 95% C.L.). This upper limit can be used to place
constraints on the strength of primordial magnetic fields, $B_{1 \rm Mpc} < 17
{\rm nG} $ (95% C.L.), and on the coupling constant of the Chern-Simons
electromagnetic term $g_{a\gamma} < 4.0 \times 10^{-2}/H_I $ (95% C.L.), where
$H_I$ is the inflationary Hubble scale. For the first time, we also
cross-correlate the CMB temperature fluctuations with the reconstructed
rotation angle map, a signal expected to be non-vanishing in certain
theoretical scenarios, and find no detectable signal. We perform a suite of
systematics and consistency checks and find no evidence for contamination.

Cosmology and Nongalactic Astrophysics;High Energy Physics - Phenomenology