  In Quantum Field Theory models with spontaneously broken gauge invariance,
renormalizability limits to four the degree of the Higgs potential, whose
minima determine the vacuum state at tree-level. In many models, this bound has
the intriguing consequence of preventing the observability, at tree-level, of
some phases that would be allowed by symmetry. We show that, generally, the
phenomenon persists also if one-loop radiative corrections are taken into
account. The tree-level unobservability of some phases is characteristic in
two-Higgs-doublet extensions of the Standard Model with additional discrete
symmetries (to protect against neutral current flavor changing effects, for
instance). We show that an extension of the scalar sector through suitable
singlet fields can resolve the {\em unnatural} limitations on the observability
of all the phases allowed by symmetry.
