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The intermediate polar FO Aquarii (FO Aqr) experienced its first-reported low-accretion states in 2016, 2017, and 2018, and using newly available photographic plates, we identify pre-discovery low states in 1965, 1966, and 1974. The primary focus of our analysis, however, is an extensive set of time-series photometry obtained between 2002 and 2018, with particularly intensive coverage of the 2016-2018 low states. After computing an updated spin ephemeris for the white dwarf (WD), we show that its spin period began to increase in 2014 after having spent 27 years decreasing; no other intermediate polar has experienced a sign change of its period derivative, but FO Aqr has now done so twice. Our central finding is that the recent low states all occurred shortly after the WD began to spin down, even though no low states were reported in the preceding quarter-century, when it was spinning up. Additionally, the system's mode of accretion is extremely sensitive to the mass-transfer rate, with accretion being almost exclusively disk-fed when FO Aqr is brighter than V~14 and substantially stream-fed when it is not. Even in the low states, a grazing eclipse remains detectable, confirming the presence of a disk-like structure (but not necessarily a Keplerian accretion disk). We relate these various observations to theoretical predictions that during the low state, the system's accretion disk dissipates into a non-Keplerian ring of diamagnetic blobs. Finally, a new XMM-Newton observation from a high state in 2017 reveals an anomalously soft X-ray spectrum and diminished X-ray luminosity compared to pre-2016 observations., Comment: Accepted for publication in ApJ

In 2016 May, the intermediate polar FO~Aqr was detected in a low state for the first time in its observational history. We report time-resolved photometry of the system during its initial recovery from this faint state. Our data, which includes high-speed photometry with cadences of just 2 sec, shows the existence of very strong periodicities at 22.5 min and 11.26 min, equivalent to the spin-orbit beat frequency and twice its value, respectively. A pulse at the spin frequency is also present but at a much lower amplitude than is normally observed in the bright state. By comparing our power spectra with theoretical models, we infer that a substantial amount of accretion was stream-fed during our observations, in contrast to the disk-fed accretion that dominates the bright state. In addition, we find that FO~Aqr's rate of recovery has been unusually slow in comparison to rates of recovery seen in other magnetic cataclysmic variables, with an $e$-folding time of 115$\pm7$ days. The recovery also shows irregular variations in the median brightness of as much as 0.2~mag over a 10-day span. Finally, we show that the arrival times of the spin pulses are dependent upon the system's overall brightness., Comment: Accepted for publication at ApJ. The paper now includes ASAS-SN observations of what we believe is the initial decline from the high state. New co-authors have been added