Several inequivalent thermodynamic formulations of spacetimes with NUT charge obey their own first laws and Smarr relations, so macroscopic consistency alone does not fix the black hole state space. We introduce a two-horizon sector diagnostic whose temperatures are fixed by harmonic-mean relations rather than adjusted as potentials. For uncharged Taub-NUT, the sum sector closes with the mass and NUT charge, whereas the difference sector closes, within a restricted homogeneous diagnostic class tested here, only after including the rotation-like thermodynamic secondary hair $J_n=mn$. Thus, within this sector diagnostic, the NUT parameter enters through both $N=n$ and the homogeneous combination $J_n=mn$. The secondary hair is defined off shell in the homogeneous mass representation, not as a new metric parameter or asymptotic charge. Hidden conformal symmetry of the rotating Kerr-NUT parent gives a consistency check, without assuming a holographic dual.
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