In the present paper we investigated the optical properties of a stack of right- and left- hand chiral photonic crystal layers. The problem is solved by Ambartsumian's layer addition modified method. We investigated the reflection spectra peculiarities of this system and showed that in contrast to a single cholesteric liquid crystal (CLC) layer this system has multiple photonic band gaps (PBGs) (at light normal incidence). We showed that this system has unique polarization properties, particularly the eigenpolarizations (EPs) of the system are degenerated (i.e., the two EPs coincide) for an even number of layers and, in contrast to ordinary gyrotropic systems, the polarization plane rotation decreases if the system thickness is increased, the rotation sign depends on the first sublayer chirality sign, the system is very sensitive to the change of the sublayer number in the system, etc. We also investigated the influence of sublayer thicknesses, incidence angle, the sublayer local dielectric anisotropies, the sublayer helix pitches on the reflection peculiarities, and other optical parameters of the system.