cmd ' (ctrl ' on win/linux) will cycle the first quote pair found (from the start/end of the section) through the following sequence:
Any other files: " to '
Also available different start and end of wrap characters: </>, [/], «/» etc. (thanks to @dirondin )
IMPORTANT USABILITY NOTE: By design, what is selected will be ignored by the toggle logic. This allows dead-simple code and maximum user flexibility to select what needs to be excluded. So the toggle will start to find the matching from the start and end of the selection.
Customizable, see below.