We analyze the past 67,000 years of climate using Antarctic ice-core records to constrain the mechanisms involved in (a) the "bipolar seesaw" relationship between Greenland and Antarctic surface temperature variations, and (b) mechanisms of millennial-scale atmospheric CO₂ concentration variations. Specifically, we determine for each Greenland Stadial the rate of Antarctic temperature and atmospheric CO₂ rise. We find that Antarctic warming rates significantly decrease as the climate cools during the glacial period, whereas the rate of atmospheric CO₂ rise does not significantly change. Also, we find that the rates of Antarctic warming and atmospheric CO₂ rise are both insensitive to whether a given stadial contains a Heinrich event. These results challenge the view that a single Southern-Ocean-based mechanism dominates the observed glacial variability in Antarctic temperature and atmospheric CO₂. Instead, our results are consistent with an important contribution of low- and mid-latitude processes to millennial-scale atmospheric CO₂ changes.