  The cosmic microwave background (CMB) photons that scatter off free electrons
in the large-scale structure induce a linear polarization pattern proportional
to the remote CMB temperature quadrupole observed in the electrons' rest frame.
The associated blackbody polarization anisotropies are known as the polarized
Sunyaev Zel'dovich (pSZ) effect. Relativistic corrections to the remote
quadrupole field give rise to a non-blackbody polarization anisotropy
proportional to the square of the transverse peculiar velocity field; this is
the kinetic polarized Sunyaev Zel'dovich (kpSZ) effect. In this paper, we
forecast the ability of future CMB and galaxy surveys to detect the kpSZ
effect, finding that a statistically significant detection is within the reach
of planned experiments. We further introduce a quadratic estimator for the
square of the peculiar velocity field based on a galaxy survey and CMB
polarization. Finally, we outline how the kpSZ effect is a probe of cosmic
birefringence and primordial non-Gaussianity, forecasting the reach of future
experiments.
