At LOFAR we measure the radio emission from extensive air showers (EAS) in the frequency band of 30 - 80 MHz in dual-polarized antennas. Through an accurate antenna calibration we can determine the complete set of four Stokes parameters that uniquely determine the linear and circular polarization of the radio signal for an EAS. The observed dependency of the circular polarization on azimuth angle and distance to the shower axis is explained as due to the interfering contributions from the two different radiation mechanisms, a main contribution due to a geomagnetically-induced transverse current and a secondary component due to the Askaryan effect. The measured data show a quantitative agreement with microscopic CORSIKA/CoREAS calculations. Having a very detailed understanding of radio emission from EAS, opens the possibility to use circular polarization as an investigative tool in the analysis of air shower structure, such as for the determination of atmospheric electric fields.