  We update and summarize the present status of our understanding of the
reparametrization symmetry with $i \leftrightarrow j$ state exchange in
neutrino oscillation in matter. We introduce a systematic method called
``Symmetry Finder'' (SF) to uncover such symmetries, demonstrate its efficient
hunting capability, and examine their characteristic features. Apparently they
have a local nature: The 1-2 and 1-3 state exchange symmetries exist at around
the solar- and atmospheric-resonances, respectively, with the level-crossing
states exchanged. However, this view is not supported, to date, in the globally
valid Denton et al. (DMP) perturbation theory, which possesses the 1-2 exchange
symmetry but not the 1-3. It is probably due to lack of our understanding, and
we find a clue for a larger symmetry structure than that we know. In the latter
part of this article, we introduce non-unitarity, or unitarity violation (UV),
into the $\nu$SM neutrino paradigm, a low-energy description of beyond $\nu$SM
new physics at high (or low) scale. Based on the analyses of UV extended
versions of the atmospheric-resonance and the DMP perturbation theories, we
argue that the reparametrization symmetry has a diagnostics capability for the
theory with the $\nu$SM and UV sectors. A speculation is given on the
topological nature of the identity which determines the transformation property
of the UV $\alpha$ parameters.
