The optical afterglow of gamma-ray burst GRB 020813 was observed for 3 hr with the LRIS spectropolarimeter at the Keck I telescope, beginning 4.7 hr after the burst was detected by the HETE-2. The spectrum reveals numerous metal absorption lines that we identify with two systems at z=1.223 and z=1.255. We also detect an [O II] łambda3727 emission line at z=1.255, and we identify this galaxy as the likely host of the GRB. After a correction for Galactic interstellar polarization, the optical afterglow has a linear polarization of 1.8%-2.4% during 4.7-7.9 hr after the burst. A measurement of p=0.80%+/-0.16% on the following night by Covino et al. demonstrates significant polarization variability over the next 14 hr. The lack of strong variability in the position angle of linear polarization indicates that the magnetic field in the jet is likely to be globally ordered rather than composed of a number of randomly oriented cells. Within the framework of afterglow models with collimated flows, the relatively low observed polarization suggests that the magnetic field components perpendicular and parallel to the shock front are only different by about 20%.