  WASP-80b is a missing link in the study of exo-atmospheres. It falls between
the warm Neptunes and the hot Jupiters and is amenable for characterisation,
thanks to its host star's properties. We observed the planet through transit
and during occultation with Warm Spitzer. Combining our mid-infrared transits
with optical time series, we find that the planet presents a transmission
spectrum indistinguishable from a horizontal line. In emission, WASP-80b is the
intrinsically faintest planet whose dayside flux has been detected in both the
3.6 and 4.5 $\mu$m Spitzer channels. The depths of the occultations reveal that
WASP-80b is as bright and as red as a T4 dwarf, but that its temperature is
cooler. If planets go through the equivalent of an L-T transition, our results
would imply this happens at cooler temperatures than for brown dwarfs. Placing
WASP-80b's dayside into a colour-magnitude diagram, it falls exactly at the
junction between a blackbody model and the T-dwarf sequence; we cannot discern
which of those two interpretations is the more likely. Flux measurements on
other planets with similar equilibrium temperatures are required to establish
whether irradiated gas giants, like brown dwarfs, transition between two
spectral classes. An eventual detection of methane absorption in transmission
would also help lift that degeneracy.
  We obtained a second series of high-resolution spectra during transit, using
HARPS. We reanalyse the Rossiter-McLaughlin effect. The data now favour an
aligned orbital solution and a stellar rotation nearly three times slower than
stellar line broadening implies. A contribution to stellar line broadening,
maybe macroturbulence, is likely to have been underestimated for cool stars,
whose rotations have therefore been systematically overestimated. [abridged]
