  [Abridged] Distant stars and planets will remain spatially unresolved for the
foreseeable future. It is nonetheless possible to infer aspects of their
brightness markings and viewing geometries by analyzing disk-integrated
rotational and orbital brightness variations. We compute the harmonic
lightcurves, F_l^m(t), resulting from spherical harmonic maps of intensity or
albedo, Y_l^m(theta,phi), where l and m are the total and longitudinal order.
Notably, odd m>1 are present in an inclined lightcurve, but not seen by an
equatorial observer. We therefore suggest that the Fourier spectrum of a
thermal lightcurve may be sufficient to determine the orbital inclination of
non-transiting short-period planets, the rotational inclination of stars and
brown dwarfs, and the obliquity of directly imaged planets. In the best-case
scenario of a nearly edge-on geometry, measuring the m=3 mode of a star's
rotational lightcurve to within a factor of two provides an inclination
estimate good to +/- 6 degrees, assuming stars have randomly distributed spots.
Alternatively, if stars have brightness maps perfectly symmetric about the
equator, their lightcurves will have no m=3 power, regardless of orientation.
In general, inclination estimates will remain qualitative until detailed
hydrodynamic simulations and/or occultation maps can be used as a calibrator.
We further derive harmonic reflected lightcurves for tidally-locked planets;
these are higher-order versions of the well-known Lambert phase curve. We show
that a non-uniform planet may have an apparent albedo 25% lower than its
intrinsic albedo, even if it exhibits precisely Lambertian phase variations.
Lastly, we provide low-order analytic expressions for harmonic lightcurves that
can be used for fitting observed photometry; as a general rule, edge-on
solutions cannot simply be scaled by sin(i) to mimic inclined lightcurves.
