Studying the resolved stellar populations of the different structural components that build massive galaxies directly unveils their assembly history. We aim at characterizing the stellar population properties of a representative sample of bulges and pure spheroids in massive galaxies (M-star > 1010 M-circle dot) in the GOODS-N field. We take advantage of the spectral and spatial information provided by SHARDS and Hubble Space Telescope data to perform the multi-image spectrophotometric decoupling of the galaxy light. We derive the spectral energy distribution separately for bulges and disks in the redshift range 0.14 < z <= 1 with spectral resolution R similar to 50. Analyzing these spectral energy distributions, we find evidence of a bimodal distribution of bulge formation redshifts. We find that 33% of them present old mass-weighted ages, implying a median formation redshift z(form) = 6.2(-1.7)(+1.5). They are relics of the early universe embedded in disk galaxies. A second wave, dominant in number, accounts for bulges formed at median redshift z(form) = 1.3(-0.6)(+0.6). The oldest (first-wave) bulges are more compact than the youngest. Virtually all pure spheroids (i.e., those without any disk) are coetaneous with the second-wave bulges, presenting a median redshift of formation z(form) = 1.1(-0.3)(+0.3). The two waves of bulge formation are distinguishable not only in terms of stellar ages but also in star formation mode. All first-wave bulges formed fast at z similar to 6, with typical timescales around 200 Myr. A significant fraction of the second-wave bulges assembled more slowly, with star formation timescales as long as 1 Gyr. The results of this work suggest that the centers of massive disk-like galaxies actually harbor the oldest spheroids formed in the universe.