We report new polarization measurements of the variable near-infrared emission of the SgrA* counterpart associated with the massive 3-4*106M_o Black Hole at the Galactic Center. We investigate the physical processes responsible for the variable emission from SgrA*. The observations have been carried out using the NACO adaptive optics (AO) instrument at the European Southern Observatory's Very Large Telescope. We find that the variable NIR emission of SgrA* is highly polarized and consists of a contribution of a non- or weakly polarized main flare with highly polarized sub-flares. The flare activity shows a possible quasi-periodicity of 20+/-3 minutes consistent with previous observations. The highly variable and polarized emission supports that the NIR emission is non-thermal. The observations can be interpreted in a jet or temporary disk model. In the disk model the quasi-periodic flux density variations can be explained due to spots on relativistic orbits around the central MBH. Alternative explanations for the high central mass concentration involving boson or fermion balls are increasingly unlikely.