Studying the resolved stellar populations of the different structural components which 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 ($M_{star}>10^{10}$ M$_{odot}$) in the GOODS-N field. We take advantage of the spectral and spatial information provided by SHARDS and HST data to perform the multi-image spectro-photometrical decoupling of the galaxy light. We derive the spectral energy distribution separately for bulges and disks in the redshift range $0.14<zleq1$ with spectral resolution $Rsim50$. Analyzing these SEDs, we find evidences of a bimodal distribution of bulge formation redshifts. We find that 33% of them present old mass-weighted ages, implying a median formation redshift $z_{rm{form}}={6.2}_{-1.7}^{+1.5}$. They are relics of the early Universe embedded in disk galaxies. A second wave, dominant in number, accounts for bulges formed at median redshift $z_{rm{form}}={1.3}_{-0.6}^{+0.6}$. The oldest (1$^{rm{st}}$-wave) bulges are more compact than the youngest. Virtually all pure spheroids (i.e., those without any disk) are coetaneous with the 2$^{rm{nd}}$-wave bulges, presenting a median redshift of formation $z_{rm{form}}={1.1}_{-0.3}^{+0.3}$. The two waves of bulge formation are not only distinguishable in terms of stellar ages, but also in star formation mode. All 1$^{rm st}$-wave bulges formed fast at $zsim6$, with typical timescales around 200 Myr. A significant fraction of the 2$^{rm{nd}}$-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.