With the arctic ice shrinking and the antarctic ice growing, isn't that going to cause some response in the earth's movement through space? It seems that any mental model I can construct of the gravitational pull between the earth and sun, moon, etc. must be perturbed.

Sorry if you've been down this street before, but am pretty new here and don't recall seeing anything.

This asymmetric build up of polar ice was at the heart of Charles Hapgood's theory of earth crustal displacement, but have no clue whether later studies have supported this. Independent ideas on the gravitational pulls would be a good starting point.