Mapping the chromosomal rearrangements between species can inform our understanding of genome evolution, reproductive isolation, and speciation. Here, we present a novel algorithm for identifying regions of synteny in pairs of genetic maps, which is implemented in the accompanying R package syntR. The syntR algorithm performs as well as previous ad hoc methods while being systematic, repeatable, and applicable to mapping chromosomal rearrangements in any group of species. In addition, we present a systematic survey of chromosomal rearrangements in the annual sunflowers, which is a group known for extreme karyotypic diversity. We build high-density genetic maps for two subspecies of the prairie sunflower, Helianthus petiolaris ssp. petiolaris and H. petiolaris ssp. fallax Using syntR, we identify blocks of synteny between these two subspecies and previously published high-density genetic maps. We reconstruct ancestral karyotypes for annual sunflowers using those synteny blocks and conservatively estimate that there have been 7.9 chromosomal rearrangements per million years, a high rate of chromosomal evolution. Although the rate of inversion is even higher than the rate of translocation in this group, we further find that every extant karyotype is distinguished by between one and three translocations involving only 8 of the 17 chromosomes. This nonrandom exchange suggests that specific chromosomes are prone to translocation and may thus contribute disproportionately to widespread hybrid sterility in sunflowers. These data deepen our understanding of chromosome evolution and confirm that Helianthus has an exceptional rate of chromosomal rearrangement that may facilitate similarly rapid diversification.