Recent experiments revealed a striking scaring behavior of the low and ultralow interfacial tension of microemulsions. A description of this behavior based on the Helfrich elastic free energy, which is symmetric in the principal curvatures c(1) and c(2), appears to be inconsistent. We show that, within the phenomenological theory of membrane bending elasticity, symmetry breaking between the two principal curvatures seems to be required in order to explain the low, but nonzero, values of the interfacial tension and its temperature dependence. We propose two simple generalizations of the Helfrich free energy which describe the experimental results. The first considers a quadratic elastic free energy and anisotropy in the membrane which breaks the symmetry between the two principal curvatures. In the second, which is applicable to systems with positive saddle-splay rigidities, the symmetry between the two principal curvatures is spontaneously broken by inclusion of higher-order terms in the curvatures in order to stabilize the free energy of the system. This analysis provides a straightforward method to obtain estimates of the bending elastic constants from interfacial tension measurements. Experiments confirming the theoretical picture are presented and values for kappa and <(kappa)over bar>, for a variety of systems, are obtained.