Paleoclimate variations occur at various time scales, between a few centuries for the Heinrich events and several hundreds of millenia for the glacial to interglacial variations. The recent ice cores from Greenland (Greenland Ice Core Project and Greenland Ice Sheet Project 2) and Antarctica (Vostok) span at least one glacial oscillation and provide many opportunities to investigate climate variations with a very fine resolution. The joint study of cores from both hemispheres allows us to distinguish between the sources of variability and helps to propose mechanisms of variations for the different time scales involved. The climate proxies we analyze are inferred from δ18O and δD for temperature and chemical species (such as calcium) for the joint behavior of the major ions in the atmosphere, which yield an estimate of the polar circulation index. Those data provide time series of climatic variables from which we extract the information on the dynamics of the underlying system. We used several independent spectral analysis techniques, to reduce the possibility of spurious results. Those methods encompass the multitaper spectral analysis, singular-spectrum analysis, maximum entropy method, principal component analysis, minimum bias spectral estimates, and digital filter reconstructions. Our results show some differences between the two hemispheres in the slow variability associated with the astronomical forcing. Common features found in the three ice-core records occur on shorter periods, between 1 and 7 kyr. The Holocene also shows recurrent common patterns between Greenland and Antarctica. We propose and discuss mechanisms to explain such behavior.