The origin of the 11-and 22-year solar cycles remains one of the more mysterious aspects of the Sun. These cycles are probably driven by convection in the solar interior, but the convection zone is difficult to probe. Small departures from sphericity in the effective surface temperature of the Sun can in principle be used in this regard. Such variations, which are observed as changes in the surface brightness with solar latitude, may be caused by differences between the vertical and horizontal turbulent convective flows  inside the Sun. Moreover, variations in the Sun's luminosity may be related to changes in conditions near the base of the convection zone  that result from the magnetic (sunspot) cycle . Here we present satellite data that show that the Sun's shape and temperature vary with latitude in an unexpectedly complex way. Although the solar oblateness shows no evidence of varying with the solar cycle, we find a significant hexadecapole shape term which may vary. We also see a variation of about 1.5 K in the surface temperature with latitude. Based on these results, we suggest that sensitive observations of brightness variations be used as a record of the surface 'shadow' of cyclical changes in the solar interior.