  The $(i)$ reciprocity relations for the relative Fisher information (RFI,
hereafter) and $(ii)$ a generalized RFI-Euler theorem, are self-consistently
derived from the Hellmann-Feynman theorem. These new reciprocity relations
generalize the RFI-Euler theorem and constitute the basis for building up a
mathematical Legendre transform structure (LTS, hereafter), akin to that of
thermodynamics, that underlies the RFI scenario. This demonstrates the
possibility of translating the entire mathematical structure of thermodynamics
into a RFI-based theoretical framework. Virial theorems play a prominent role
in this endeavor, as a Schr\"odinger-like equation can be associated to the
RFI. Lagrange multipliers are determined invoking the RFI-LTS link and the
quantum mechanical virial theorem. An appropriate ansatz allows for the
inference of probability density functions (pdf's, hereafter) and
energy-eigenvalues of the above mentioned Schr\"odinger-like equation. The
energy-eigenvalues obtained here via inference are benchmarked against
established theoretical and numerical results. A principled theoretical basis
to reconstruct the RFI-framework from the FIM framework is established.
Numerical examples for exemplary cases are provided.
