DNA base damage is a major source of oncogenic mutations1. Such damage can produce strand-phased mutation patterns and multiallelic variation through the process of lesion segregation2. Here, we exploited these properties to reveal how strand-asymmetric processes, such as replication and transcription, shape DNA damage and repair. Despite distinct mechanisms of leading and lagging strand replication3,4, we observe identical fidelity and damage tolerance for both strands. For small DNA adducts, our results support a model in which the same translesion polymerase is recruited on-the-fly to both replication strands, starkly contrasting the strand asymmetric tolerance of bulky adducts5. We find that DNA damage tolerance is also common during transcription, where RNA-polymerases frequently bypass lesions without triggering repair. At multiple genomic scales, we show the pattern of DNA damage induced mutations is largely shaped by the influence of DNA accessibility on repair efficiency, rather than gradients of DNA damage. Finally, we reveal specific genomic conditions that can corrupt the fidelity of nucleotide excision repair and actively drive oncogenic mutagenesis. These results provide insight into how strand-asymmetric mechanisms underlie the formation, tolerance, and repair of DNA damage, thereby shaping cancer genome evolution.Competing Interest StatementP.F. is a member of the Scientific Advisory Boards of Fabric Genomics, Inc. and Eagle Genomics, Ltd. The other authors declare no competing interests.