Quantification of the velocity and magnetic field reversals in dynamo remains an interesting challenge. In this paper, using group-theoretic analysis, we classify the reversing and non-reversing Fourier modes during a dynamo reversal in a Cartesian box. Based on odd-even parities of the wavenumber indices, we categorise the velocity and magnetic Fourier modes into eight classes each. Then, using the properties of the nonlinear interactions in magnetohydrodynamics, we show that these 16 elements form Klein 16-group Z2×Z2×Z2×Z2. We demonstrate that field reversals in a class of Taylor-Green dynamo, as well as the reversals in earlier experiments and models, belong to one of the classes predicted by our group-theoretic arguments.