Clumped isotopic measurements of nitrous oxide (N₂O) have the potential to offer unique constraints on the processes governing N₂O production and destruction, building on the information provided by δ¹⁵N, δ¹⁸O and ¹⁵N site preference (SP). Extending their application requires a robust absolute reference frame. Here we show that thermal equilibration of N₂O over γ-Al₂O₃ provides such a reference frame for measurements of the isotopologues ¹⁴N¹⁵N¹⁸O and ¹⁵N¹⁴N¹⁸O, as well as for SP. Using a quantum cascade laser absorption spectroscopy (QCLAS) platform, we simultaneously quantify seven isotopologues of N₂O, including ¹⁴N¹⁵N¹⁸O, ¹⁵N¹⁴N¹⁸O, and ¹⁵N¹⁵N¹⁶O. Experiments starting from isotopically-distinct starting materials show convergence to time-invariant compositions that are in agreement with theoretically predicted temperature dependencies. These results demonstrate that γ-Al₂O₃ activated at ≥ 550 °C catalyzes isotope exchange among isotopologues of N₂O at equilibration temperatures between 153 °C and 218 °C and thereby define an absolute stochastic reference frame for Δ¹⁴N¹⁵N¹⁸O and Δ¹⁵N¹⁴N¹⁸O. In contrast, ¹⁵N¹⁵N¹⁶O does not equilibrate under these conditions, suggesting selective activation of N–O but not N–N bonds. Comparison of equilibrium SP values with theoretical predictions reveals a systematic offset relative to the current reference scale, which will require future work to reconcile.