  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.
