The bit that made this click for me: it works on any commit in history, not just HEAD. Mark the commit as edit in git rebase -i, then git reset HEAD^ to uncommit it while keeping the changes in the working tree, and rebuild the pieces from there.

git add -p is the real workhorse for the rebuild: s splits a hunk into smaller ones, and e lets you hand-edit the hunk when the boundary doesn't fall on clean line breaks. Stage a coherent slice, git commit, repeat until the tree is empty, then git rebase --continue. The rebase's only job is to drop you at the right spot; add -p does the actual splitting.