Don't think this one is in the catalog of official refactorings but one that would be a time saver for me is the ability to split a class from source + header file with other classes into a source and header file just for that class. This one is fresh on my memory since I inherited a legacy C++ project that had 50+ sets of files with multiple classes. Fortunately I found out about Source Monitor which was a big help in tracking those files down but only after I had found about half of them on my own. Even then SM reported some files as having 12+ classes in them. Still had to split the classes out manually. Kept wishing that VAX could have done a lot of the mechanical stuff for me. Seems doable since VAX can move implementation back and forth already.
For now using VA Outline should help, since you can select the items you want in VA Outline, cut them and then paste them into the new code files. This does require making the new code files manually, but it should still help a bit.