Almost never happens with indexes. Vols rise with underlying falling. It behaves more like sticky strike. Almost never happens - vols fall with underlying rising. Same as stocks. Not that different. Index is just a few stocks traded in one basket.
My understanding was that the IV skew is the same for stock indexes and stocks. i.e. IV is higher for lower strikes than the corresponding (same number of strikes OTM) higher strikes. This is due to perceived risks of price crashes in the stock market. Commodities are what have the opposite skew. i.e. higher IV for higher strikes than corresponding lower strikes. This is due to perceived risk of price spikes due to supply crashes like a corn crop failure for example. In both cases IV is higher in the direction of perceived higher risk where people insure their long underlying positions with options.
Actually, most of the index skew is due to the correlation pickup at lower strike. Here is a simple example - ATM top 50 implied correlation for 1 month is in the low 50s while var-swap correlation is in the high 50s. So the skews in single stocks is obviously lower, if you measure it on consistent basis.