Polarized and unpolarized neutron diffractions have been carried out to investigate the nature of the magnetic structures and transitions in monoclinic Co3TeO6. As the temperature is lowered below 26 K long range order develops, which is fully incommensurate (ICM) in all three crystallographic directions. Below 19.5 K additional commensurate magnetic peaks develop, consistent with the T-4 irreducible representation, along with a splitting of the ICM peaks along the h direction which indicates that there are two separate sets of magnetic modulation vectors. Below 18 K, this small additional magnetic incommensurability disappears, ferroelectricity develops, an additional commensurate magnetic structure consistent with T-3 irreducible representation appears, and the k component of the ICM wave vector disappears. Synchrotron x-ray diffraction measurements demonstrate that there is a significant shift of the electronic charge distribution from the Te ions at the crystallographic 8f sites to the neighboring Co and O ions. These results, together with the unusually small electric polarization, its strong magnetic field dependence, and the negative thermal expansion in all three lattice parameters, suggest this material is an antiferroelectric. Below15 K the k component of the ICM structure reappears, along with second-order ICM Bragg peaks, which polarized neutron data demonstrate are magnetic in origin.