Mutational processes constantly shape the somatic genome, leading to immunity, aging, cancer, and other diseases. When cancer is the outcome, we are afforded a glimpse into these processes by the clonal expansion of the malignant cell. Here, we characterize a less explored layer of the mutational landscape of cancer: mutational asymmetries between the two DNA strands. Analyzing whole-genome sequences of 590 tumors from 14 different cancer types, we reveal widespread asymmetries across mutagenic processes, with transcriptional ("T-class") asymmetry dominating UV-, smoking-, and liver-cancer-associated mutations and replicative ("R-class") asymmetry dominating POLE-, APOBEC-, and MSI-associated mutations. We report a striking phenomenon of transcription-coupled damage (TCD) on the non-transcribed DNA strand and provide evidence that APOBEC mutagenesis occurs on the lagging-strand template during DNA replication. As more genomes are sequenced, studying and classifying their asymmetries will illuminate the underlying biological mechanisms of DNA damage and repair.