We investigate precision observables sensitive to custodial-symmetric/-violating UV physics beyond the Standard Model. We use the Standard Model effective field theory (SMEFT) framework which in general includes nonoblique corrections that require a generalization of the Peskin-Takeuchi $T$ parameter to unambiguously detect custodial symmetry/violation. We take a first step towards constructing a SMEFT reparametrization-invariant replacement, which we call $𝒯$, valid at least for tree-level custodial-violating contributions. We utilize a new custodial basis of $\nu \mathrm{SMEFT}$ (SMEFT augmented by right-handed neutrinos) which explicitly identifies the global $SU(2{)}_R$ symmetries of the Higgs and fermion sectors, which 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 $𝒯$ parameter at tree level. We illustrate our results with several UV theory examples, demonstrating that $𝒯$ faithfully identifies custodial symmetry violation, while $T$ can fail.