While gravitational microlensing by planetary systems$^1,2$ provides unique vistas on the properties of exoplanets$^3$, observations of a given two-body microlensing event can often be interpreted with multiple distinct physical configurations. Such ambiguities are typically attributed to the close-wide$^4,5$ and inner-outer$^6$ types of degeneracy, which arise from transformation invariances and symmetries of microlensing caustics. However, there remain unexplained inconsistencies (see, for example, ref. $^7$) between the aforementioned theories and observations. Here, leveraging a fast machine learning inference framework$^8$, we present the discovery of the offset degeneracy, which concerns a magnification-matching behaviour on the lens axis and is formulated independently of caustics. This offset degeneracy unifies the close-wide and inner-outer degeneracies, generalizes to resonant topologies and, upon reanalysis, not only appears ubiquitous in previously published planetary events with twofold degenerate solutions, but also resolves prior inconsistencies. Our analysis demonstrates that degenerate caustics do not strictly result in degenerate magnifications and that the commonly invoked close- wide degeneracy essentially never arises in actual events. Moreover, it is shown that parameters in offset-degenerate configurations are related by a simple expression. This suggests the existence of a deeper symmetry in the equations governing two-body lenses than previously recognized.