  We have discovered a \sim420d modulation, with associated X-ray dips, in
RXTE-ASM/MAXI/Swift-BAT archival light-curves of the short-period (3.2h)
black-hole X-ray transient, Swift J1753.5-0127. This modulation only appeared
at the end of a gradual rebrightening, approximately 3 years after the initial
X-ray outburst in mid-2005. The same periodicity is present in both the 2-20
keV and 15-50 keV bands, but with a \sim0.1 phase offset (\sim40d).
Contemporaneous photometry in the optical and near-IR reveals a weaker
modulation, but consistent with the X-ray period. There are two substantial
X-ray dips (very strong in the 15-50 keV band, weaker at lower energies) that
are separated by an interval equal to the X-ray period. This likely indicates
two physically separated emitting regions for the hard X-ray and lower energy
emission. We interpret this periodicity as a property of the accretion disc,
most likely a long-term precession, where the disc edge structure and X-ray
irradiation is responsible for the hard X-ray dips and modulation, although we
discuss other possible explanations, including Lense-Thirring precession in the
inner disc region and spectral state variations. Such precession indicates a
very high mass ratio LMXB, which even for a \sim10M_sun BH requires a brown
dwarf donor (\sim0.02M_sun), making Swift J1753.5-0127 a possible analogue of
millisecond X-ray pulsars.We compare the properties of Swift J1753.5-0127 with
other recently discovered short-period transients, which are now forming a
separate population of high latitude BH transients located in the galactic
halo.
