Can I hard link files with rsync instead of copying them?

The answer has turned out to be yes, with rsync's --link-dest option. It's just not obvious because it's not a simple on/off flag like cp -l, because it takes a path argument.

So, cp -l a/ b/ can also be done like this:

rsync  -a  --link-dest=../a  a/  b/

Note that the destination needs to be ../a, because the path is relative to the relative to the target directory.
(So if you were syncing to b/c/, it would be ../../a.)

I've been using https://github.com/rsnapshot/rsnapshot for a long time. It does exactly what you are describing.