We have found that there is a significant difference in the intensity-time profiles of the galactic cosmic rays with energies from 100 MeV to 50 GeV around the solar activity minima in the five alternate solar magnetic field polarities from 1954 to 2000. This difference in the appearance of the two halves of the 22-year solar magnetic cycle, peaked and flat-topped, supports the role of drifts near the cosmic ray intensity maxima. The intensity of cosmic rays at the solar activity minima also depends on the solar magnetic field polarity, and it is less at neutron monitor energies when the solar magnetic field polarity is positive than when it is negative. At energies of ~500 MeV and lower the opposite is the case: The values of the intensity maxima are less when the solar magnetic field polarity is negative. This intensity difference arises from the role of drifts in the cosmic ray propagation, together with possibly a change in the value of the diffusion coefficient in the two polarity cycles, that produces changes in the shape of the proton and helium energy spectra. The rigidity dependence of the ratios of the maximum intensities of both protons and helium nuclei below 1 and 2 GV, respectively, in alternate solar magnetic cycles (e.g., 1987/1977) is given by P(0.45¿0.05), but the ratios are shifted by a factor of 2 in rigidity. This shift does not appear to be predicted by current modulation theories that include cosmic ray drift effects. ¿ 2001 American Geophysical Union