  Phase curve measurements provide a global view of the composition, thermal
structure, and dynamics of exoplanet atmospheres. Although most of the dozens
of phase curve measurements made to date are of large, massive hot Jupiters,
there is considerable interest in probing the atmospheres of the smaller
planets that are the more typical end product of planet formation. One such
planet is the ultra-hot Neptune LTT 9779b, a rare denizen of the Neptune
desert. A companion paper presents the planet's secondary eclipses and day-side
thermal emission spectrum; in this work we describe the planet's optical and
infrared phase curves, characterized using Spitzer and TESS photometry. We
detect LTT 9779b's thermal phase variations at 4.5um, finding a phase amplitude
of 358+/-106 ppm and a longitude of peak emission -10 deg +/- 21 deg east of
the substellar point. Combined with our secondary eclipse observations, these
phase curve measurements imply a 4.5um day-side brightness temperature of
1800+/-120 K, a night-side brightness temperature of 700+/-430 K (<1350 K at 2
sigma confidence), and a day-night brightness temperature contrast of
1110+/-460 K. We compare our data to the predictions of 3D GCMs and to similar
observations of hot Jupiters experiencing similar levels of stellar
irradiation. Though not conclusive, our measurement of its small 4.5um phase
offset, the relatively large amplitude of the phase variation, and the
qualitative differences between our target's day-side emission spectrum and
those of hot Jupiters of similar temperatures all suggest a super-Solar
atmospheric metallicity for LTT 9779b, as might be expected given its size and
mass. Finally, we provide a refined ephemeris (P=0.79207022+/-0.00000069 d,
T0=2458783.51636+/-0.00027, BJD_TDB) to enable efficient scheduling of future
observations to further characterize the atmosphere of this intriguing planet.
(abstract abridged)
