  We investigate precision observables sensitive to custodial
symmetric/violating UV physics beyond the Standard Model. We use the SMEFT
framework which in general includes non-oblique corrections that requires a
generalization of the Peskin-Takeuchi $T$ parameter to unambiguously detect
custodial symmetry/violation. We take a first step towards constructing a SMEFT
reparameterization-invariant replacement, that we call $\mathscr{T}$, valid at
least for tree-level custodial violating contributions. We utilize a new
custodial basis of $\nu$SMEFT (SMEFT augmented by right-handed neutrinos) which
explicitly identifies the global $SU(2)_R$ symmetries of the Higgs and fermion
sectors, that in turn permits easy identification of higher-dimensional
operators that are custodial preserving or violating. We carefully consider
equation-of-motion redundancies that cause custodial symmetric operators in one
basis to be equivalent to a set of custodial symmetric and/or violating
operators in another basis. Utilizing known results about tree/loop operator
generation, we demonstrate that the basis-dependent appearance of
custodial-violating operators does not invalidate our $\mathscr{T}$ parameter
at tree-level. We illustrate our results with several UV theory examples,
demonstrating that $\mathscr{T}$ faithfully identifies custodial symmetry
violation, while $T$ can fail.
