Protein synthesis by the ribosome involves large-scale rearrangements of the "small" subunit (SSU; ~ 1 MDa), which include inter- and intra-subunit rotational motions. With more than 1000 structures of ribosomes and ribosomal subunits now publicly available, it is becoming increasingly difficult to design precise experiments that are based on a comprehensive analysis of all known rotation states. To overcome this limitation, we present the Ribosome Angle Decomposition (RAD) method, where the orientation of each small subunit head and body is described in terms of three angular coordinates (rotation, tilt and tilt direction) and a single translation. To demonstrate the utility of the accompanying software (RADtool) we applied it to all published ribosome and mitoribosome structures. This identified and analyzed 1077 fully-assembled ribosome complexes, as well as 280 isolated small subunits from 48 organisms. The RAD approach quantitatively distinguishes between previously described qualitative rotational features, determines when rotation-only descriptions are insufficient, and shows that tilt-like rearrangements of the SSU head and body are pervasive in both prokaryotic and eukaryotic ribosomes. Together, the presented database and technique provide a robust platform for systematically analyzing, visualizing, and comparing subunit orientations of ribosomes from all kingdoms of life. Accordingly, the RAD resource establishes a common foundation with which structural, simulation, single-molecule and biochemical efforts can precisely interrogate the dynamics of this prototypical molecular machine.