With hardlinks, does it matter which is which? If I make replace duplicate files with hardlinks within my home (~) directory, I'm thinking it is completely irrelevant which of two files gets replaced.

Is this true, or is there some asymmetry that I am not aware of?
No - hard links work like individual files. As long as you make the next link/copy in the correct folder/directory - you are good to go. When you cross a filesystem boundary - the link breaks automagically and it becomes a distinct file.