A tantalizing enigma in extragalactic astronomy concerns the chronology and driving mechanisms of the buildup of the central luminosity excess (bulge) in spiral galaxies like our Milky Way, i.e. in systems referred to as late-type galaxies (LTGs) in the Hubble classification scheme. The standard scenario envisages a two-phase galaxy formation process, with the bulge assembling first in a quick and violent quasi-monolithic episode, with the disk gradually building up around it. The morphological outcome of this inside-out galaxy formation process is a bulge made of old, red stars that is surrounded by a younger disk whose appearance is dominated by blue massive stars and ionized gas. However, in sharp contrast to the fossil classical bulges (CBs) predicted by this scenario, most present-day LTGs host bluish bulges that exhibit low-level ongoing star formation (SF). These, so-called, pseudo-bulges (PBs) are thought to emerge from a second, ’slow’ route to bulge formation along secular galactic evolution, primarily through gentle gas inflow from the disk feeding in its center quasi-continuous SF and the gradual growth of a bulge-like luminosity excess. These two scenarios, the first one envisaging bulge formation prior to disks and the second one out of disks, obviously yield antipodal predictions on the bulge age and the bulge-to-disk color contrast, both expected to be high (low) in the first (second) case. A combined analysis of integral field spectroscopy data from the CALIFA survey and imaging data from SDSS for a representative sample of LTGs in the nearby universe shows that galaxy bulges form over three orders of magnitude in mass a continuous sequence of increasing age. This relation rules out an evolutionary dichotomy between CBs and PBs in local LTGs, and argues against the hypothesis of PBs being SF-rejuvenated CBs.