The spatiotemporal organization of eukaryotic DNA is controlled by an intricate combination of passive (thermally activated) and active (ATP-consuming) processes. Based on recent experimental insights into the mechanochemical cycle of chromatin remodelers, molecular machines that actively control nucleosome positions, we introduce a model to study the competition between active and passive mechanisms at the most basic layer of DNA packaging, the wrapping of DNA into nucleosomes. Depending on the level of remodeler activity, the positions of nucleosomes are controlled by either the bending or the stretching energy of the wrapped DNA involved. Since these energies are highly sequence dependent, DNA guides its own packaging. However, since this dependence differs for the two deformation modes, active processes can drive nucleosomes from their equilibrium positions. Furthermore, for repetitive DNA sequences, such as telomeres, we find thermal ratchets that propel nucleosomes in a preferred direction.