Whole-genome duplications are radical evolutionary events that have driven speciation and adaptation in many taxa. Higher-order polyploids have complex histories often including interspecific hybridization and subsequent dynamic genomic changes. This chromosomal reshuffling and restructuring is poorly understood for most polyploid species, despite their evolutionary and agricultural importance, due to the challenge of distinguishing homologous sequences from each other. Here we use dense linkage maps generated with targeted sequence capture to improve the diploid strawberry (Fragaria vesca) reference genome and to disentangle the subgenomes of the wild octoploid progenitors of cultivated strawberry, F. virginiana and F. chiloensis. Our novel approach, POLiMAPS (Phylogenetics Of Linkage-Map-Anchored Polyploid Subgenomes) leverages sequence reads to associate informative inter-homeolog phylogenetic markers with linkage groups and reference genome positions. Using POLiMAPS, we uncover three major results: First, in contrast to a widely accepted model, we find that one of the four subgenomes originates with the diploid cytoplasm donor F. vesca, one with the diploid F. iinumae, and two with an unknown ancestor close to F. iinumae. Second, we find extensive unidirectional gene conversion changing F. iinumae-like subgenomes to be more F. vesca-like, but never the reverse, suggesting that selection favors either nuclear sequence compatible with the F. vesca-derived cytoplasm, or else rapid divergence between similar subgenomes to facilitate disomic inheritance. Third, we find evidence of numerous interchromosomal rearrangements, suggesting that transpositions may contribute substantially to divergence between homeologous chromosomes. These results highlight the complicated web of genetic exchanges that occur during polyploid evolution and suggest a path forward for unraveling other polyploid genomes.