DNA supercoiling is a fundamental biological process occurring in all cells. We developed a theory of braiding (supercoiling) of a pair of DNA molecules that takes into account the contribution of the bending and the electrostatic energy. The electrostatic interaction was calculated within the framework of the Kornyshev-Leikin theory of DNA interactions, which takes into account realistic helical patterns of charge. Because of the chirality of the charge patterns, we predict that left-handed braiding of a pair of DNA molecules is more favourable than right-handed braiding. Applying our model to the case of closed loop DNA supercoiling and to single molecule DNA micromanipulations, we predict novel effects that have not yet been experimentally observed. We show that supercoiling may occur in topologically relaxed plasmids, as a consequence of attractive chiral forces. We speculate about the potential biological role of the predicted effects in the case of topoisomerase action, and the occurrence of positively supercoiled DNA in hyperthermophilic bacteria and archea. Our findings also suggest alternative an explanation of well-known experiments that proved that divalent ions overwind DNA. We also give an explanation for pairing of homologous DNA molecules in monovalent salt, and explain the occurrence of tight supercoiling observed in cryo-electron and atomic force microscopy. The analysis of existing experimental data shows that in most cases the chiral effects that we predict remain elusive. The theory therefore awaits final experimental verification.