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Difference-in-differences (DiD) is a popular method to evaluate treatment effects of real-world policy interventions. Several approaches have previously developed under alternative identifying assumptions in settings where pre- and post-treatment outcome measurements are available. However, these approaches suffer from several limitations, either (i) they only apply to continuous outcomes and the average treatment effect on the treated, or (ii) they depend on the scale of the outcome, or (iii) they assume the absence of unmeasured confounding given pre-treatment covariate and outcome measurements, or (iv) they lack semiparametric efficiency theory. In this paper, we develop a new framework for causal identification and inference in DiD settings that satisfies (i)-(iv), making it universally applicable, unlike existing DiD methods. Key to our framework is an odds ratio equi-confounding (OREC) assumption, which states that the generalized odds ratio relating treatment and treatment-free potential outcome is stable across pre- and post-treatment periods. Notably, the framework recovers the standard DiD model under a certain simple location-shift model, but readily generalizes to nonlinear scales. Under the OREC assumption, we establish nonparametric identification for any potential treatment effect on the treated in view, which in principle would be identifiable under the stronger assumption of no unmeasured confounding. Moreover, we develop a consistent, asymptotically linear, and semiparametric efficient estimator of treatment effects on the treated by leveraging recent learning theory. We illustrate our framework through simulation studies and two real-world applications using Zika virus outbreak data and traffic safety data.