Using Sitecore 8.2 Update 3
I had the following folder structure in templates:
-User Defined
--Global
--User Defined
--|--Core
--|--Site Specific
--|--|--Site 1
--|--|--Site 2
I did some reorganization, so now my Master database looks like this:
-User Defined
--Core
--User Defined
--|--Site Specific
--|--|--Site 1
--|--|--Site 2
When I published (republish) it seemed to move everything around OK, but it wont rename the Global folder to Core, so my web actually looks like this:
-User Defined
--Global
--User Defined
--|--Site Specific
--|--|--Site 1
--|--|--Site 2
In another instance, I changed the template of an item, and that does not seem to want to publish either.