Nucleosomes form the unit structure of the genome in eukaryotes, thereby constituting a fundamental tenet of chromatin biology. In canonical nucleosomes, DNA wraps around the histone octamer in a left-handed toroidal ramp. Here, in single-molecule magnetic tweezers studies of chaperone-assisted nucleosome assembly, we show that the handedness of the DNA wrapping around the nucleosome core is intrinsically ambidextrous, and depends on the pre-assembly supercoiling state of the DNA, i.e., it is not uniquely determined by the octameric histone core. Nucleosomes assembled onto negatively supercoiled DNA are found to exhibit a left-handed conformation, whereas assembly onto positively supercoiled DNA results in right-handed nucleosomes. This intrinsic flexibility to adopt both chiralities is observed both for canonical H3 nucleosomes, and for centromere-specific variant CENP-A nucleosomes. These data support recent advances suggesting an intrinsic adaptability of the nucleosome, and provide insights into how nucleosomes might rapidly re-assemble after cellular processes that generate positive supercoiling in vivo.