[This was posted as a comment on Lucia’s Blackboard blog but disappeared down a borehole. It is repeated here but may disappear if the original comment shows up]
No, I don’t have any references because I did not look for any. However, I can give you some visual examples of this unequal variability property.
For a while now, I have been studying daily station series and it seems quite clear that winter temperatures are more variable than summer ones and that this effect is very strongly accentuated as one moves toward the poles. Recently BEST published their newly minted temperature series and at their web page, they produced a gridded equal-area cell construction of 5498 monthly land temperature series.
I took a time-truncated subset of all of these series starting in February, 1956 (the date was chosen because it was the earliest date from which all of the cells had values for each month) and continuing to the most recent values. For each series, the standard deviation of the temperature was calculated separately for each month.
The first pair of plots gives a plot of the SD by latitude for the months of February and August. The red lines are references for the tropics.
The second pair is a plot versus longitude with point from the northern and southern hemisphere indicated by color.
Admittedly, I have not detrended the anomalies first, but the climate variation for a cell should not be large enough to create such large differences in the SDs between months.
Unequal variability would create a fairly substantial stumbling block for finding changepoints in series.