In budding yeast, a single cenH3 (Cse4) nucleosome occupies the ∼120-bp functional centromere, however conflicting structural models for the particle have been proposed. To resolve this controversy, we have applied H4S47C-anchored cleavage mapping, which reveals the precise position of histone H4 in every nucleosome in the genome. We find that cleavage patterns at centromeres are unique within the genome and are incompatible with symmetrical structures, including octameric nucleosomes and (Cse4/H4)2 tetrasomes. Centromere cleavage patterns are compatible with a precisely positioned core structure, one in which each of the 16 yeast centromeres is occupied by oppositely oriented Cse4/H4/H2A/H2B hemisomes in two rotational phases within the population. Centromere-specific hemisomes are also inferred from distances observed between closely-spaced H4 cleavages, as predicted from structural modeling. Our results indicate that the orientation and rotational position of the stable hemisome at each yeast centromere is not specified by the functional centromere sequence. DOI: http://dx.doi.org/10.7554/eLife.01861.001.