Measures of climate are known to exhibit scaling behavior with large exponents, resulting in larger fluctuations at longer timescales. It has been suggested that climate models underestimate these fluctuations [1-4], casting doubt on their ability to predict the amplitude of climate variability over coming decades and centuries. Using the latest simulations and data syntheses, as well as spectral methods tailored to scaling estimation, we find agreement for spectra derived from observations and models on timescales ranging from interannual to multi-millennial. Our results confirm the existence of a regime transition between orbital and annual peaks [5], occurring around millennial periodicities. That both simple and comprehensive ocean-atmosphere models can reproduce these features suggests that long-range persistence is a consequence of the oceanic integration of both gradual and abrupt climate forcings. The result implies that decadal to millennial variability over the Holocene is partly a consequence of the climate system’s integrated memory of orbital forcing. While climate models imperfectly depict some aspects of spatiotemporal variability, we find that they appear contain the essential physics to correctly simulate the temperature continuum. We hypothesize that the deep ocean plays a key role in integrating forcings, keeping a long memory of past events, and having the ability to strongly influence climate states. We therefore suggest that a critical element of successful simulations at sub-orbital scales are initial conditions of the deep ocean state that are consistent with observations of the recent past. Failing to provide such initial conditions sets the models up for failure.