Cancer genomes often harbor hundreds of somatic DNA rearrangement junctions, many of which cannot be easily classified into simple (e.g., deletion) or complex (e.g., chromothripsis) structural variant classes. Applying a novel genome graph computational paradigm to analyze the topology of junction copy number (JCN) across 2,778 tumor whole-genome sequences, we uncovered three novel complex rearrangement phenomena: pyrgo, rigma, and tyfonas. Pyrgo are “towers” of low-JCN duplications associated with early-replicating regions, superenhancers, and breast or ovarian cancers. Rigma comprise “chasms” of low-JCN deletions enriched in late-replicating fragile sites and gastrointestinal carcinomas. Tyfonas are “typhoons” of high-JCN junctions and fold-back inversions associated with expressed protein-coding fusions, breakend hypermutation, and acral, but not cutaneous, melanomas. Clustering of tumors according to genome graph-derived features identified subgroups associated with DNA repair defects and poor prognosis.