  The work within the Geometrothermodynamics programme rests upon the metric
structure for the thermodynamic phase-space. Such structure exhibits discrete
Legendre symmetry. In this work, we study the class of metrics which are
invariant along the infinitesimal generators of Legendre transformations. We
solve the Legendre-Killing equation for a $K$-contact general metric. We
consider the case with two thermodynamic degrees of freedom, i.e. when the
dimension of the thermodynamic phase-space is five. For the generic form of
contact metrics, the solution of the Legendre-Killing system is unique, with
the sole restriction that the only independent metric function -- $\Omega$ --
should be dragged along the orbits of the Legendre generator. We revisit the
ideal gas in the light of this class of metrics. Imposing the vanishing of the
scalar curvature for this system results in a further differential equation for
the metric function $\Omega$ which is not compatible with the Legendre
invariance constraint. This result does not allow us to use the regular
interpretation of the curvature scalar as a measure of thermodynamic
interaction for this particular class.
