  Physical observables cannot depend on the basis one chooses to describe
fields. Therefore, all physically relevant properties of a model are, in
principle, expressible in terms of basis-invariant combinations of the
parameters. However, in many cases it becomes prohibitively difficult to
establish key physical features exclusively in terms of basis invariants. Here,
we advocate an alternative route in such cases: the formulation of
basis-invariant statements in terms of basis-covariant objects. We give several
examples where the basis-covariant path is superior to the traditional approach
in terms of basis invariants. In particular, this includes the formulation of
necessary and sufficient basis-invariant conditions for various physically
distinct forms of CP conservation in two- and three-Higgs-doublet models.
