We study the cosmological evolution of asymmetries in the two-Higgs doublet extension of the Standard Model, prior to the electroweak phase transition. If Higgs flavour-exchanging interactions are sufficiently slow, then a relative asymmetry among the Higgs doublets corresponds to an effectively conserved quantum number. Since the magnitude of the Higgs couplings depends on the choice of basis in the Higgs doublet space, we attempt to formulate basis-independent out-of-equilibrium conditions. We show that an initial asymmetry between the Higgs scalars, which could be generated by CP violation in the Higgs sector, will be transformed into a baryon asymmetry by the sphalerons, without the need of $B-L$ violation. This novel mechanism of baryogenesis through (split) Higgsogenesis is exemplified with simple scenarios based on the out-of-equilibrium decay of heavy singlet scalar fields into the Higgs doublets.