Computer simulations of the geodynamo reveal a strong inverse correlation between stability and equatorial symmetry of the simulated field. This result is consistent with the paleomagnetically inferred degree of symmetry of Earth's magnetic field during the past 150 Ma, which reversed polarity more frequently when it was more symmetrical. A geodynamo simulation with solid inner core much smaller than it is today produced an exceptionally antisymmetric field, raising the possibility that reversals may have been much less common in the distant geologic past than more recently. Paleomagnetic evidence suggests that this might well be true.