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Because all climate models exhibit biases, their use for assessing future climate change requires implicitly assuming or explicitly postulating that the biases are stationary or vary predictably. This hypothesis, however, has not been, and cannot be, tested directly. This work shows that under very large climate change the bias patterns of key climate variables exhibit a striking degree of stationarity. Using only correlation with a model's preindustrial bias pattern, a model's 4xCO(2) bias pattern is objectively and correctly identified among a large model ensemble in almost all cases. This outcome would be exceedingly improbable if bias patterns were independent of climate state. A similar result is also found for bias patterns in two historical periods. This provides compelling and heretofore missing justification for using such models to quantify climate perturbation patterns and for selecting well-performing models for regional downscaling. Furthermore, it opens the way to extending bias corrections to perturbed states, substantially broadening the range of justified applications of climate models.