For what it's worth, my opinion is that when the two words in question have different meanings, then both should be allowed.
An obvious example is pants (an item of clothing) which is allowed even though it is identical in form to pants (= 'gasps').
Both pant and pants are allowed as common.
Here, axes (plural of 'axis') is quite obviously a different word from the plural of axe, so both should (imho) be allowed.
It even has a different pronunciation. (Sorry, I don't know how to do phonetic symbols on the forum)
(This particular question is further complicated by the fact that axes is also the plural of ax, which is how some Americans spell the word!)
However, since Alan has declared a moratorium on dealing with all questions relating to plurals, the matter is purely academic.