Polyploidy often confers emergent properties, such as the higher fibre productivity and quality of tetraploid cottons than diploid cottons bred for the same environments¹. Here we show that an abrupt five- to sixfold ploidy increase approximately 60 million years (Myr) ago, and allopolyploidy reuniting divergent Gossypium genomes approximately 1–2 Myr ago², conferred about 30–36-fold duplication of ancestral angiosperm (flowering plant) genes in elite cottons (Gossypium hirsutum and Gossypium barbadense), genetic complexity equalled only by Brassica³ among sequenced angiosperms. Nascent fibre evolution, before allopolyploidy, is elucidated by comparison of spinnable-fibredGossypium herbaceum A and non-spinnable Gossypium longicalyx F genomes to one another and the outgroup D genome of non-spinnable Gossypium raimondii. The sequence of a G. hirsutum AtDt (in which ‘t’ indicates tetraploid) cultivar reveals many non-reciprocal DNA exchanges between subgenomes that may have contributed to phenotypic innovation and/or other emergent properties such as ecological adaptation by polyploids. Most DNA-level novelty in G. hirsutum recombines alleles from the D-genome progenitor native to its New World habitat and the Old World A-genome progenitor in which spinnable fibre evolved. Coordinated expression changes in proximal groups of functionally distinct genes, including a nuclear mitochondrial DNA block, may account for clusters of cotton-fibre quantitative trait loci affecting diverse traits. Opportunities abound for dissecting emergent properties of other polyploids, particularly angiosperms, by comparison to diploid progenitors and outgroups.