Photonic bandgap design is one of the most basic ways to effectively control the interaction between light and matter. However, the traditional photonic bandgap is always dispersive (blueshift with the increase of the incident angle), which is disadvantageous to the construction of wide-angle optical devices. Hypercrystal, the photonic crystal with layered hyperbolic metamaterials (HMMs), can strongly modify the bandgap properties based on the anomalous wavevector dispersion of the HMM. Here, based on phase variation competition between HMM and isotropic dielectric layers, we propose for the first time to design nonreciprocal and flexible photonic bandgaps in one-dimensional photonic crystals containing magneto-optical HMMs. Especially the zero-shift cavity mode and the blueshift cavity mode are designed for the forward and backward propagations, respectively. Our results show maximum absorption about 0.99 (0.25) in an angle range of 20-75 degrees for the forward (backward) incident light at the wavelength of 367 nm. The nonreciprocal omnidirectional cavity mode not only facilitates the design of perfect unidirectional optical absorbers working in a wide-angle range, but also possesses significant applications for all-angle reflectors and filters.