SummaryCancer genomes often harbor hundreds of somatic DNA rearrangement junctions, many of which cannot be easily classified into simple (e.g. deletion, translocation) or complex (e.g. chromothripsis, chromoplexy) structural variant classes. Applying a novel genome graph computational paradigm to analyze the topology of junction copy number (JCN) across 2,833 tumor whole genome sequences (WGS), we introduce three complex rearrangement phenomena: pyrgo, rigma, and tyfonas. Pyrgo are “towers” of low-JCN duplications associated with early replicating regions and superenhancers, and are enriched in breast and ovarian cancers. Rigma comprise “chasms” of low-JCN deletions at late-replicating fragile sites in esophageal and other gastrointestinal (GI) adenocarcinomas. Tyfonas are “typhoons” of high-JCN junctions and fold back inversions that are enriched in acral but not cutaneous melanoma and associated with a previously uncharacterized mutational process of non-APOBEC kataegis. Clustering of tumors according to genome graph-derived features identifies subgroups associated with DNA repair defects and poor prognosis.