Close binary systems often show linear polarization varying over the binary period, usually attributed to light scattered from electrons in circumstellar clouds. One of the brightest close binary systems is Spica (Alpha Virginis) consisting of two B type stars orbiting with a period of just over 4 days. Past observations of Spica have shown low polarization with no evidence for variability. Here we report new high-precision polarization observations of Spica that show variation with an amplitude ~200 parts-per-million (ppm). Using a new modelling approach we show that the phase-dependent polarization is primarily due to reflected light from the primary off the secondary and vice versa. The stars reflect only a few per-cent of the incident light, but the reflected light is very highly polarized. The polarization results show that the binary orbit is clockwise and the position angle of the line of nodes is 130.4 +/- 6.8 degrees in agreement with Intensity Interferometer results. We suggest that reflected light polarization may be much more important in binary systems than has previously been recognized and may be a way of detecting previously unrecognized close binaries.