In this paper we extend the HEALPix discrete global grid system (DGGS) to ellipsoids of revolution, thereby broadening its possible applications. Elaborating on the work of Calabretta and Roukema [Mapping on the HEALPix grid, Monthly Notices of the Royal Astronomical Society 381 (2007), no. 2, 865872.], we also rearrange the HEALPix map projection and build a new DGGS on top of it, which we call the rHEALPix DGGS. The rHEALPix DGGS has all the key features of the HEALPix DGGS, and its planar projection consists of horizontal-vertical aligned nested square grids, which makes it easy to understand and display. We present all the formulas and algorithms necessary for a basic implementation of the rHEALPix DGGS and link to our own open source implementation thereof.