Upon discussing this idea with my peers it seems that the reflexivity axiom just means that the group of goods which we are comparing are well defined. The traditional reading of the definition being "a good is at least as good as itself" is an extremely exact way of summing up this idea.
An example of non-reflexive preferences could be explained via example:
Say you walk into a pizza store and have decided to choose a pepperoni slice of pizza out of all the variety of items on the menu , the slices that you see for sale over the counter are cut in a extremely uneven way.
Thus we can see that though we can have complete and transitive preferences, we do not find reflexivity! (since some slices are bigger than the others).
This is a result of misspecified commodity groups or inability to separate them into their own groups (organizing slices not only by type but by size as well).
Is your preference across all these slices the same?