  We revisit the issue of mixing matrix renormalization in theories that
include Dirac or Majorana fermions. We show how a gauge-variant on-shell
renormalized mixing matrix can be related to a manifestly gauge-independent one
within a generalized ${\bar {\rm MS}}$ scheme of renormalization. This
scheme-dependent relation is a consequence of the fact that in any scheme of
renormalization, the gauge-dependent part of the mixing-matrix counterterm is
ultra-violet safe and has a pure dispersive form. Employing the unitarity
properties of the theory, we can successfully utilize the afore-mentioned
scheme-dependent relation to preserve basic global or local symmetries of the
bare Lagrangian through the entire process of renormalization. As an immediate
application of our study, we derive the gauge-independent renormalization-group
equations of mixing matrices in a minimal extension of the Standard Model with
isosinglet neutrinos.
