  In magnetic crystals, despite the explicit breaking of time-reversal
symmetry, two equilibrium states related by time reversal are always
energetically degenerate. In ferromagnets such time-reversal degeneracy can be
manifested by hysteresis loops in the magnetic field dependence of the
magnetization and, if metallic, in the anomalous Hall effect. Importantly, both
quantities simply change signs but not their absolute sizes under time
reversal, which follows from their fundamental definitions. Our integral
experimental and theoretical study shows that in the metallic kagome spin ice
HoAgGe subject to finite magnetic fields parallel to the kagome plane, an
emergent time-reversal-like degeneracy appears between magnetic states that
have the same energy and net magnetization, but different sizes of the
anomalous Hall effect. These degeneracies are unraveled by finite hysteresis in
the field-dependent anomalous Hall effect contrasted with the vanishing
hysteresis in the magnetization, which appears only at low-temperatures T$<$4K
when the kagome spin ice is fully ordered into $\sqrt{3}$$\times$$\sqrt{3}$
state. By explicitly determining the degenerate states and calculating the
corresponding physical properties using a tight-binding model, we nailed down
the time-reversal-like operation that transforms these degenerate states into
each other. The operation is related to the nontrivial distortion of the kagome
lattice in HoAgGe and is effective only because of the richness of degenerate
states unique to kagome spin ice. Our work points to the powerful role of the
anomalous Hall effect to diagnose hidden symmetries in frustrated spin systems.
