Polyploidy often confers emergent properties, such as the higher fibre productivity and quality of tetraploid cottons than diploid cottons bred for the same environments. 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 ago, 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 Brassica among sequenced angiosperms. Nascent fibre evolution, before allopolyploidy, is elucidated by comparison of spinnable-fibred Gossypium 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 A(t)D(t) (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.