To understand the Active Galactic Nuclei (AGN) phenomenon and their impact on the evolution of galaxies, a complete AGN census is required; however, finding heavily obscured AGNs is observationally challenging. Here we use the deep and extensive multi-wavelength data in the COSMOS field to select a complete sample of 578 infrared (IR) quasars (LAGN,IR>1045ergs−1) at z < 3, with minimal obscuration bias, using detailed UV-to-far-IR spectral energy distribution (SED) fitting. We complement our SED constraints with X-ray and radio observations to further investigate the properties of the sample. Overall, 322 of the IR quasars are detected by Chandra and have individual X-ray spectral constraints. From a combination of X-ray stacking and L2−10keV – L6μm analyses, we show that the majority of the X-ray faint and undetected quasars are heavily obscured (many are likely Compton thick), highlighting the effectiveness of the mid-IR band to find obscured AGNs. We find that 355 (≈ 61 per cent) IR quasars are obscured (NH>1022cm−2) and identify differences in the average properties between the obscured and unobscured quasars: (1) obscured quasars have star-formation rates ≈3 times higher than unobscured systems for no significant difference in stellar mass and (2) obscured quasars have stronger radio emission than unobscured systems, with a radio-loudness parameter ≈0.2dex higher. These results are inconsistent with a simple orientation model but in general agreement with either extreme host-galaxy obscuration towards the obscured quasars or a scenario where obscured quasars are an early phase in the evolution of quasars.