Aims:The “bright” High Frequency Peakers (HFPs) sample is a mixture of blazars and intrinsically small and young radio sources. We investigate the polarimetric characteristics of 45 High Frequency Peakers, from the “bright” HFP sample, in order to have a deeper knowledge of the nature of each object, and to construct a sample made of genuine young radio sources only. Methods: Simultaneous VLA observations carried out at 22.2, 15.3, 8.4 and 5.0 GHz, together with the information at 1.4 GHz provided by the NVSS at an earlier epoch, have been used to study the linearly polarized emission. Results: From the analysis of the polarimetric properties of the 45 sources we find that 26 (58%) are polarized at least at one frequency, while 17 (38%) are completely unpolarized at all frequencies. We find a correlation between fractional polarization and the total intensity variability. We confirm that there is a clear distinction between the polarization properties of galaxies and quasars: 17 (66%) quasars are highly polarized, while all the 9 galaxies are either unpolarized (<0.2%) or marginally polarized with fractional polarization below 1%. This suggests that most HFP candidates identified with quasars are likely to represent a radio source population different from young radio objects.