We develop a concept of parallel existence of the ordinary (O) and hidden (H) worlds. We compare two cases: (1) when the hidden sector of the Universe is a mirror counterpart of the ordinary world, and (2) when it is a superstring-inspired shadow world described, in contrast to the mirror world, by a symmetry group (or by a chain of groups), which does not coincide with the ordinary world symmetry group. We construct a cosmological model assuming the existence of the superstring-inspired $E_6$ unification, broken at the early stage of the Universe to $SO(10)\times U(1{)}_Z$—in the O-world, and to $SU(6{)}^{\prime }\times SU(2{)}_{\theta }^{\prime }$—in the H-world. As a result, we obtain the low-energy symmetry group $G_{\mathrm{SM}}^{\prime }\times SU(2{)}_{\theta }^{\prime }$ in the shadow world, instead of the standard model group $G_{\mathrm{SM}}$ existing in the O-world. The additional non-Abelian $SU(2{)}_{\theta }^{\prime }$ group with massless gauge fields, ”thetons,” is responsible for dark energy. Considering a quintessence model of cosmology with an inflaton $\sigma$ and an axion $a_{\theta }$, which is a pseudo Nambu-Goldstone boson induced by the $SU(2{)}_{\theta }^{\prime }$-group anomaly, we explain the origin of dark energy, dark matter and ordinary matter. In the present model we review all cosmological epochs (inflation, reheating, recombination and nucleosynthesis), and give our version of the baryogenesis. The cosmological constant problem is also briefly discussed. DOI: http://dx.doi.org/10.1103/PhysRevD.84.063510 © 2011 American Physical Society