ABSTRACT The properties of the stellar populations in a galaxy are known to correlate with the amount and the distribution of stellar mass. We take advantage of the maps of light-weighted mean stellar age $\mathit {\mathrm{ Age}_{\mathrm{ wr}}}$ and metallicity $\mathit {\mathrm{ Z}_{*\mathrm{ wr}}}$ for a sample of 362 galaxies from the integral-field spectroscopic survey CALIFA (summing up to $\gt 600\, 000$ individual regions of ∼1 kpc linear size), produced in our previous works, to investigate how these local properties react to the local stellar-mass surface density μ* and to the global total stellar mass M* and mean stellar-mass surface density 〈μ*〉e. We establish the existence of (i) a dual μ*–$\mathit {\mathrm{ Age}_{\mathrm{ wr}}}$ relation, resulting in a young sequence and an old ridge, and (ii) a μ*–$\mathit {Z_{*wr}}$ relation, overall independent of the age of the regions. The global mass parameters (M* and, possibly secondarily, 〈μ*〉e) determine the distribution of μ* in a galaxy and set the maximum attainable μ*, which increases with M*. M* affects the shape and normalization of the local relations up to a threshold mass of $\sim 10^{10.3}\, \mathrm{M}_\odot$, above which they remain unchanged. We conclude that stellar mass is a ‘glocal’ (i.e. simultaneously global and local) driver of the stellar population properties. We consider how the local and global mass–age and mass–metallicity relations are connected, and in particular discuss how it is possible, from a single local relation, to produce two different global mass–metallicity relations for quiescent and star-forming galaxies, respectively, as reported in the literature. Structural differences in these two classes of galaxies are key to explain the duality in global scaling relations and appear as essential in modelling the baryonic cycle of galaxies.