Mammalian karyotypes (number and structure of chromosomes) can vary dramatically over short evolutionary time frames. There are examples of massive karyotype conversion, from mostly telocentric (centromere terminal) to mostly metacentric (centromere internal), in 10(2)-10(5) years. These changes typically reflect rapid fixation of Robertsonian (Rb) fusions, a common chromosomal rearrangement that joins two telocentric chromosomes at their centromeres to create one metacentric. Fixation of Rb fusions can be explained by meiotic drive: biased chromosome segregation during female meiosis in violation of Mendel's first law. However, there is no mechanistic explanation of why fusions would preferentially segregate to the egg in some populations, leading to fixation and karyotype change, while other populations preferentially eliminate the fusions and maintain a telocentric karyotype. Here we show, using both laboratory models and wild mice, that differences in centromere strength predict the direction of drive. Stronger centromeres, manifested by increased kinetochore protein levels and altered interactions with spindle microtubules, are preferentially retained in the egg. We find that fusions preferentially segregate to the polar body in laboratory mouse strains when the fusion centromeres are weaker than those of telocentrics. Conversely, fusion centromeres are stronger relative to telocentrics in natural house mouse populations that have changed karyotype by accumulating metacentric fusions. Our findings suggest that natural variation in centromere strength explains how the direction of drive can switch between populations. They also provide a cell biological basis of centromere drive and karyotype evolution.