In the standard cosmological model, the assembly of galaxies is primarily driven by the growth of their host dark matter halos. At the centre of these halos, however, baryonic processes take over, leading to the plethora of observed galaxy properties. The coupling between baryonic and dark matter physics is central to our understanding of galaxies, and yet, it remains a challenge for theoretical models and observations. Here, we demonstrate that measured ages, metallicities, stellar angular momentum, morphology and star formation rates are correlated with both stellar and halo mass. We find using dynamical modelling that at fixed stellar mass, CALIFA galaxies are younger, more metal-poor and rotationally supported and have higher star formation rates and later-type morphologies as their total mass increases, due to the independent measurements of the stellar and total masses. These results indicate that the formation of galaxies and, thus, their baryonic properties do not vary with stellar mass alone, with halo mass also playing an important role.