I found the following page in the ArcGIS javascript samples that supposedly gives an example of how to store tiles locally:
https://developers.arcgis.com/javascript/jssamples/exp_webstorage.html
It says that it doesn't work with IE - and I tested it and they are right - it doesn't work with IE.
Does anybody know of an approach to storing tiles locally that does work with IE? In other words, is this really a problem with IE or is the example just not robust enough?