Anticrossing behavior between magnons in the noncollinear chiral magnet Cu2OSeO3 and a two-mode X-band microwave resonator was studied in the temperature range 5-100 K. In the field-induced ferrimagnetic phase, we observed a strong-coupling regime between magnons and two microwave cavity modes with a cooperativity reaching 3600. In the conical phase, cavity modes are dispersively coupled to a fundamental helimagnon mode, and we demonstrate that the magnetic phase diagram of Cu2OSeO3 can be reconstructed from the measurements of the cavity resonance frequency. In the helical phase, a hybridized state of a higher-order helimagnon mode and a cavity mode-a helimagnon polariton-was found. Our results reveal a class of magnetic systems where strong coupling of microwave photons to nontrivial spin textures can be observed.