To ascertain whether photometric decompositions of galaxies into bulges and discs are astrophysically meaningful, we have developed a new technique to decompose spectral data cubes into separate bulge and disc components, subject only to the constraint that they reproduce the conventional photometric decomposition. These decompositions allow us to study the kinematic and stellar population properties of the individual components and how they vary with position, in order to assess their plausibility as discrete elements, and to start to reconstruct their distinct formation histories. An initial application of this method to Calar Alto Integral Field Area integral field unit observations of three isolated S0 galaxies confirms that in regions where both bulge and disc contribute significantly to the flux, they can be physically and robustly decomposed into a rotating dispersion-dominated bulge component and a rotating low-dispersion disc component. Analysis of the resulting stellar populations shows that the bulges of these galaxies have a range of ages relative to their discs, indicating that a variety of processes are necessary to describe their evolution. This simple test case indicates the broad potential for extracting from spectral data cubes the full spectral data of a wide variety of individual galaxy components, and for using such decompositions to understand the interplay between these various structures, and hence how such systems formed.