Some recent observations seem to disagree with hierarchical theories of galaxy formation about the role played by major mergers in the late buildup of massive E-S0s. We re-address this question by analysing the morphology, structural distortion level and star formation enhancement of a sample of massive galaxies (M* > 5 × 1010 M&sun;) lying on the Red Sequence and its surroundings at 0.3 < z < 1.5. We have used an initial sample of ˜1800 sources with Ks < 20.5 mag over an area ˜155 arcmin2 on the Groth Strip, combining data from the Rainbow Extragalactic Database and the Galaxy Origins and Young Assembly survey. Red galaxy classes that can be directly associated with intermediate stages of major mergers and with their final products have been defined. We report observational evidence of the existence of a dominant evolutionary path among massive red galaxies at 0.6 < z < 1.5, consisting in the conversion of irregular discs into irregular spheroids, and of these ones into regular spheroids. This result implies: (1) the massive red regular galaxies at low redshifts derive from the irregular ones populating the Red Sequence and its neighbourhood at earlier epochs up to z ˜ 1.5; (2) the progenitors of the bulk of present-day massive red regular galaxies have been discs that seem to have migrated to the Red Sequence mostly through major mergers at 0.6 < z < 1.2 (these mergers thus starting at z ˜ 1.5) and (3) the formation of E-S0s that end up with M* > 1011 M&sun; at z = 0 through gas-rich major mergers has frozen since z ˜ 0.6. All these facts support that major mergers have played a dominant role in the definitive buildup of present-day E-S0s with M* > 1011 M&sun; at 0.6 < z < 1.2, in good agreement with hierarchical scenarios of galaxy formation.