<p>We classify real order-two principal symbols of classical pseudodifferential operators acting on symmetric covariant two-tensors under one-sided longitudinal gauge degeneracy and metric-only frozen-symbol isotropy, without assuming self-adjointness. The general symbol has three invariant components: scalar actions on the transverse–traceless and transverse-trace summands, and a canonical nilpotent coupling from the transverse trace line to the longitudinal scalar line. This third component is the only obstruction to two-sided longitudinal degeneracy; it vanishes in particular for a fiberwise self-adjoint principal symbol. The familiar two-coefficient transverse–traceless plus transverse-trace formula is therefore a corollary rather than the general classification. We also prove that local Weyl invariance at a critical metric annihilates the transverse trace mode and hence yields a pure transverse–traceless principal symbol, provided the pseudodifferential realization and isotropy hypotheses hold. Finally, an explicit Ricci-dependent natural symbol shows that naturality does not imply metric-only frozen-symbol isotropy. The results isolate a precise representation-theoretic symbol class and do not purport to classify all natural geometric Hessians.</p>