Abstract Context: Galactic cosmic ray (GCR) intensities around solar minimum times are modulated by magnetic drifts that depend on the overall solar polarity. GCR intensities reach a higher but more narrow peak during negative minima than during positive minima. However, despite these higher intensities, the variation of GCRs over timescales of solar rotation is smaller during negative minima than during positive minima. Aims: We study the variation of GCR intensity over the 27-day synodic solar rotation and over the 14-day half-rotation, in particular the long-term trend and cyclic pattern of this variation, and propose a unifying explanation for the observations. Methods: We used two high-latitude neutron monitors, Oulu and Apatity, which are most sensitive to the low-energy part of the GCR spectrum and thereby more strongly affected by the changes in the conditions of the local heliosphere. We calculated the yearly mean amplitudes of the GCR intensity variation during the full solar rotation (A27) and half-rotation (A14) in 1964–2016 Results: We verify that the A27 and A14 amplitudes exhibit a clear 22-yr Hale cycle during solar minima at both stations, with larger amplitudes in positive minima. We find that the mean amplitude of the Hale cycle is about 30–45% of the mean amplitude for A14, while is only about 15–30% for A27. We also find that all amplitudes depict a declining long-term trend, which we suggest is due to the weakening of solar polar magnetic fields during the last four solar cycles and the ensuing latitudinal widening of the heliospheric current sheet (HCS) region. An exceptionally wide HCS region during the last solar minimum, when A14 reached its all-time minimum, is demonstrated by Ulysses probe observations. Conclusions: Our results emphasize the effect of polarity-dependent drift and the properties of the HCS in modulating the variation of GCR intensity during solar rotation in solar minimum times. The second rotation harmonic yields a larger Hale amplitude than the first because it is more probable for the Earth to be outside the HCS only once during the rotation than twice or more, which more strongly reduces A14 during negative polarity times than A27. With the HCS region widening from minimum to minimum, the decrease in A14 is relatively faster than in A27.