  We show that a large class of sets of leptonic texture zeros considered in
the literature imply the vanishing of certain CP-odd weak-basis invariants.
These invariant conditions enable one to recognize a flavour model
corresponding to a set of texture zeros, when written in an arbitrary
weak-basis where the zeros are not manifest. We also analyse the r\^ ole of
texture zeros in allowing for a connection between leptogenesis and low-energy
leptonic masses, mixing and CP violation. For some of the textures the
variables relevant for leptogenesis can be fully determined in terms of low
energy parameters and heavy neutrino masses.
