We present high-sensitivity polarimetric observations ({sigma}_P_~=0.6mJy) in six bands covering the 5.5-38GHz range of a complete sample of 53 compact extragalactic radio sources brighter than 200mJy at 20GHz. The observations, carried out with the Australia Telescope Compact Array, achieved a 91 per cent detection rate (at 5{sigma}). Within this frequency range, the spectra of about 95 per cent of sources are well fitted by double power laws, both in total intensity and in polarization, but the spectral shapes are generally different in the two cases. Most sources were classified as either steep- or peaked-spectrum but less than 50 per cent have the same classification in total and in polarized intensity. No significant trends of the polarization degree with flux density or with frequency were found. The mean variability index in total intensity of steep-spectrum sources increases with frequency for a 4-5yr lag, while no significant trend shows up for the other sources and for the 8yr lag. In polarization, the variability index, which could be computed only for the 8yr lag, is substantially higher than in total intensity and has no significant frequency dependence.