  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.
