During the past two decades, experiments in both the northern and southern hemispheres have observed a small but measurable energy-dependent sidereal anisotropy in the arrival direction distribution of Galactic cosmic rays with relative intensities at the level of one per mille. Individually, these measurements are restricted by limited sky coverage, and so the power spectrum of the anisotropy obtained from any one measurement displays a systematic correlation between different multipole modes $C_\ell$. We present the results of a joint analysis of the anisotropy on all angular scales using cosmic-ray data collected during 336 days of operation of the High-Altitude Water Cherenkov (HAWC) Observatory (located at 19° N) and 5 years of data taking from the IceCube Neutrino Observatory (located at 90° S) The results include a combined sky map and an all-sky power spectrum in the overlapping energy range of the two experiments at around 10 TeV. We describe the methods used to combine the IceCube and HAWC data, address the individual detector systematics, and study the region of overlapping field of view between the two observatories.