  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 $0.14<z\leq1$ with spectral resolution $R\sim50$. 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
$z\sim6$, 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.
