We demonstrate by theory and experiments that well-aligned cholesteric liquid crystals that function as 1D chiral photonic crystals (CPCs) having extraordinarily large-period-numbers N (N=d/Λ; d: thickness, Λ: grating period) exceeding 500 possess many optical properties that are impossible with conventional thin CPCs. Even far from the circular Bragg resonance, these CPCs are capable of simultaneously high transmission and large broadband polarization rotation of vector beams; the polarization rotation is independent of relative orientation of the input beam polarization vectors, and a good degree of linear polarization of the output beam can be maintained.