Perhaps the problem is the terminology. It is perhaps not best to use the words "natural variables." However, we do have the case that the relation: dU = Tds - PdV + udN is widely known as "the thermodynamic identity" and this is given rather high importance in thermodynamics.
The thermodynamic identity is consistent with defining a thermodynamic temperature, and thermodynamic pressure, and thermodynamic chemical potential as...
1/T = (dS/dU) (constant V and N)
P = T(dS/dV) (constant U and N)
u = -T(dS/dN) (constant U and V)
These respectively relate thermal interactions, mechanical interactions, and diffusive processes with changes in entropy. This is, I think, generally considered the starting point to relate the statistical mechanical microscopic view of entropy to the classical macroscopic thermodynamics view of entropy.
Although these relations can lead to other "identities" as John states, this set perhaps gives the clearest connection between statistical and classical entropy, and therefore might deserve some special prominence, such as being called THE thermodynamic identity.
The wording "natural variables" can disappear as far as I am concerned.
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu