We have investigated microwave nonreciprocity in the non-centrosymmetric magnet CuB2O4. We simultaneously observed nonreciprocities with different origins: the classical magnetic dipolar nonreciprocity and the magnetochiral (MCh) nonreciprocity. By rotating a magnetic field in a tetragonal plane, we clearly unveiled a qualitative difference between them. The MCh signal reveals chiral transitions from one enantiomer to the other via an intermediate achiral state. We showed that the magnetoelectric effect plays an essential role in the emergence of the microwave MCh effect. Our demonstration may give a new insight into the classical dipolar-type and emergent spin-orbit-type microwave nonreciprocities in media with broken time-reversal symmetry and spatial inversion symmetry.