Cell-cycle control is accomplished by cyclin-dependent kinases (CDKs), motivating extensive research into CDK targeting small-molecule drugs as cancer therapeutics. Here we use combinatorial CRISPR/Cas9 perturbations to uncover an extensive network of functional interdependencies among CDKs and related factors, identifying 43 synthetic-lethal and 12 synergistic interactions. We dissect CDK perturbations using single-cell RNAseq, for which we develop a novel computational framework to precisely quantify cell-cycle effects and diverse cell states orchestrated by specific CDKs. While pairwise disruption of CDK4/6 is synthetic-lethal, only CDK6 is required for normal cell-cycle progression and transcriptional activation. Multiple CDKs (CDK1/7/9/12) are synthetic-lethal in combination with PRMT5, independent of cell-cycle control. In-depth analysis of mRNA expression and splicing patterns provides multiple lines of evidence that the CDK-PRMT5 dependency is due to aberrant transcriptional regulation resulting in premature termination. These inter-dependencies translate to drug-drug synergies, with therapeutic implications in cancer and other diseases.