I took the UIUC polar ice area daily data since 2000, and calculated 60 day running means for the anomalies. In the graph above, the Arctic data is shifted horizontally to the right by six months (to normalise seasons) and the Antarctic data is shifted vertically upwards by 0.5 million km² to normalise the baselines.
There appears to be an (imperfect) symmetry. When Arctic ice anomalies decrease, the tendency is for Antarctic ice anomalies to increase. Particularly notable are the opposite record spikes in 2007/2008.
Instead of jumping into the standard ozone explanation, it would be nice to see some studies about what is really going on to cause this relationship.