Instead of atomic references, a fragment reference basis set using e.g. isodesmic reactions can be used to integrate data from various sources (DFT, accurate gas-phase ∆ f H from the ATcT, and experimental adsorption enthalpies) into a global thermochemical network. The isodesmic reference reactions of this basis set lead to an increased accuracy of ∆ f H of adsorbates by maximizing error cancellation. Finally, we show how the linear algebra framework can also be used to "invert" the problem of constructing reaction enthalpies from ∆ f H. This approach allows direct calculation of ∆ f H from tabulated reaction enthalpies, providing an alternative perspective on how thermochemical data can be converted and combined.