Tests of the concordance cold dark matter model on the scale of galaxies are so far inconclusive due to our poor understanding of the interplay between baryons and dark matter (DM). Two critical limitations in previous efforts to disentangle the baryonic and DM distributions have been the lack of (i) two-dimensional, spatially complete and radially extended kinematics to infer the total mass distribution, and (ii) coverage in wavelength to robustly constrain the baryonic mass distribution and isolate the DM contribution. Both are now provided by existing integral-field spectroscopic data from the CALIFA survey of a statistically well-defined sample of ~600 nearby galaxies of all Hubble types. We apply dynamical and stellar population modelling in a homogeneous way to the same data. In this way we for the first time constrain both the normalisation (ratio of dwarf to giant stars) and shape (single versus broken power-law slope) of the stellar initial mass function (IMF). We then robustly characterise the mass distribution of galaxies, from dwarf-star dominance at the high-mass end to dark matter excess in low-mass spirals. In this way, CALIFA yields physical insights into the baryonic and DM interplay for a statistically well-defined sample of nearby galaxies, providing in turn crucial constraints on galaxy formation and evolution models.