Most living systems rely on double-stranded DNA (dsDNA) to store their genetic information and perpetrate themselves. Thus, the biological information contained within a dsDNA molecule, in terms of a linear sequence of nucleotides, has been considered the main target of the evolution. However, in this information-centred perspective, certain DNA sequence symmetries are difficult to explain. Here we show that these patterns can emerge from the physical peculiarities of the dsDNA molecule itself and the maximum entropy principle alone, rather than from biological or environmental evolutionary pressure. Our predictions are valid for both prokaryotes and eukaryotes, and also inform the interpretation of observed codon biases and context-dependent mutation patterns in human populations. Our results suggest that the double helix energy constraints and, more generally, the physical properties of the dsDNA are the hard drivers of the overall DNA sequence architecture, whereas the biological selective processes act as soft drivers, which only under extraordinary circumstances overtake the overall entropy content of the genome.