We present a new self-consistent thermodynamic formalism for the interfacial properties of nanoscale embryos whose interiors do not exhibit bulklike behavior and are in complete equilibrium with the surrounding mother phase. In contrast to the standard Gibbsian analysis, whereby a bulk reference pressure based on the same temperature and chemical potentials of the mother phase is introduced, our approach naturally incorporates the normal pressure at the center of the embryo as an appropriate reference pressure. While the interfacial properties of small embryos that follow from the use of these two reference pressures are different, both methods yield by construction the same reversible work of embryo formation as well as consistency between their respective thermodynamic and mechanical routes to the surface tension. Hence, there is no a priori reason to select one method over another. Nevertheless, we argue, and demonstrate via a density-functional theory (with the local density approximation) analysis of embryo formation in the pure component Lennard-Jones fluid, that our new method generates more physically appealing trends. For example, within the new approach the surface tension at all locations of the dividing surface vanishes at the spinodal where the density profile spanning the embryo and mother phase becomes completely uniform (only the surface tension at the Gibbs surface of tension vanishes in the Gibbsian method at this same limit). Also, for bubbles, the location of the surface of tension now diverges at the spinodal, similar to the divergent behavior exhibited by the equimolar dividing surface (in the Gibbsian method, the location of the surface of tension vanishes instead). For droplets, the new method allows for the appearance of negative surface tensions (the Gibbsian method always yields positive tensions) when the normal pressures within the interior of the embryo become less than the bulk pressure of the surrounding vapor phase. Such a prediction, which is allowed by thermodynamics, is consistent with the interpretation that the mother phase's attempted compression of the droplet is counterbalanced by the negative surface tension, or free energy cost to decrease the interfacial area. Furthermore, for these same droplets, the surface of tension can no longer be meaningfully defined (the surface of tension always remains well defined in the Gibbsian method). Within the new method, the dividing surface at which the surface tension equals zero emerges as a new lengthscale, which has various thermodynamic analogs to and similar behavior as the surface of tension.