Reciprocity relations and generalized, classic entropic quantifiers that lack trace-form

In this effort we show that the Legendre reciprocity relations, thermodynamics' essential formal feature, are respected by general classic entropic functionals, even if they are NOT of trace-form nature, in contrast with Shannon's or Tsallis' cases. Further, with reference to the Maximum Entropy (MaxEnt) variational process, we encounter important cases, relevant to physical applications currently discussed in the research literature, in which the associated reciprocity relations exhibit anomalies. We show that these anomalies can be cured by carefully discriminating between apparently equivalent entropic forms.