  We propose an invariant see-saw (ISS) approach to model building, based on
the observation that see-saw models of neutrino mass and mixing fall into basis
invariant classes labelled by the Casas-Ibarra $R$-matrix, which we prove to be
invariant not only under basis transformations but also non-unitary
right-handed neutrino transformations $S$. According to the ISS approach, given
any see-saw model in some particular basis one may determine the invariant $R$
matrix and hence the invariant class to which that model belongs. The
formulation of see-saw models in terms of invariant classes puts them on a
firmer theoretical footing, and allows different see-saw models in the same
class to be related more easily, while their relation to the $R$-matrix makes
them more easily identifiable in phenomenological studies. We also present an
ISS mass formula which may be useful in model building. To illustrate the ISS
approach we show that sequential dominance (SD) models form basis invariant
classes in which the $R$-matrix is approximately related to a permutation of
the unit matrix, and quite accurately so in the case of constrained sequential
dominance (CSD) and tri-bimaximal mixing. Using the ISS approach we discuss
examples of models in which the mixing naturally arises (at least in part) from
the charged lepton or right-handed neutrino sectors and show that they are in
the same invariant class as SD models. We also discuss the application of our
results to flavour-dependent leptogenesis where we show that the case of a real
$R$ matrix is approximately realized in SD, and accurately realized in CSD.
