The ribosome translates the genetic code into proteins in all domains of life. Its size and complexity demand long-range interactions that regulate ribosome function. These interactions are largely unknown. Here, we apply a global coevolution method, statistical coupling analysis (SCA), to identify coevolving residue networks (sectors) within the 23S ribosomal RNA (rRNA) of the large ribosomal subunit. As in proteins, SCA reveals a hierarchical organization of evolutionary constraints with near-independent groups of nucleotides forming physically contiguous networks within the three-dimensional structure. Using a quantitative, continuous-culture-with-deep-sequencing assay, we confirm that the top two SCA-predicted sectors contribute to ribosome function. These sectors map to distinct ribosome activities, and their origins trace to phylogenetic divergences across all domains of life. These findings provide a foundation to map ribosome allostery, explore ribosome biogenesis, and engineer ribosomes for new functions. Despite differences in chemical structure, protein and RNA enzymes appear to share a common internal logic of interaction and assembly.