Solar modulation refers to Galactic cosmic ray (GCR) variations with the $\sim$11-year sunspot cycle and $\sim$22-year solar magnetic cycle, and is relevant to the space radiation environment and effects on Earth's atmosphere. Its complicated dependence on solar and heliospheric conditions is only roughly understood but has been empirically modeled in terms of a single modulation parameter. Most analyses of solar modulation used neutron monitor (NM) data from locations with relatively low geomagnetic cutoff rigidity, i.e., the threshold for cosmic rays to penetrate Earth's magnetic field. The Princess Sirindhorn Neutron Monitor (PSNM) at Doi Inthanon, Thailand has the highest cutoff rigidity ($\approx$17 GV) where observations span a complete solar modulation cycle (since late 2007). The pattern of solar modulation at Doi Inthanon during 2011-2014 was qualitatively very different from that at low geomagnetic cutoff, and is not well described by the same modulation parameter. At other times, NM count rates from Doi Inthanon and McMurdo, Antarctica (cutoff $<$0.01 GV) are linearly correlated and confirm the observation from latitude surveys in the previous solar cycle that the slope of the correlation changes with solar magnetic polarity. Low solar magnetic tilt angles ($<$40$^{\circ}$, at negative polarity) were well correlated with variations at both NM stations as expected from drift models. At higher tilt angle, the Doi Inthanon count rate is well correlated with the interplanetary magnetic field, which is consistent with an expected increase in diffusion at high rigidity short-circuiting effects of drifts and the heliospheric current sheet.