Exchanging roles of variable and thermal potential

In thermal physics entropy S is regarded as a system variable in 1st theorem of thermal dynamics, i.e.
$$dE(S,V)=TdS-pdV=T(S,V)dS-p(S,V)dV$$

Entropy is regarded also as thermal potential,i.e.

$$dS(E,V)=\frac{1}{T(E,V)}dE + \frac{p(E,V)}{T(E,V)}dV$$

S,V,p,T are usually regarded as variables. My question is that other variables V, p, T can be also treated as thermal potentials ? Can exchanging roles of variable and thermal potential take place only for the pair of E and S ?

$$dV(S,E)=\frac{T(S,E)}{p(S,E)}dS-\frac{1}{p(S,E)}dE$$
$$dT(F,V)=\frac{p(F,V)}{S(F,V)}dV-\frac{1}{S(F,V)}dF$$
$$dp(G,T)=\frac{1}{V(G,T)}dG+\frac{S(G,T)}{V(G,T)}dT$$
$$F=E-TS$$
$$G=E-TS+PV$$