I am experiencing something similar, and perhaps it is related, but does not sound like the exact same thing. For my web map, I have 4 CSV layers from the web added on top of the default Topographic basemap. Pretty straight-forward. I order them the way I want them to be, set drawing and popup info, and save my map. All looks great. If I close out of the web map (click My Content), and re-open the same web map, the layer order has been reversed. If I change NOTHING in the map, and just save it again and re-open it, the layers are reversed again. This opening and saving has become my (increasingly annoying) work-around for this bug (can we agree that it's a bug??). I have built a JavaScript application that consumes this web map and depends on the order of layers for some of the data presentation, but since the configuration order keeps changing unexpectedly, my application keeps behaving unexpectedly. Please help!
Thanks,
Alan
Stone Environmental
Montpelier, VT