I strongly agree with your advice to do code tidying as a separate commit from any substantive changes. I often tidy as I work on code, but using Mercurial and TortoiseHg I can cherry pick commits so it is generally easy to pick up all the "tidying" related changes and commit those then preform the "substantive changes" commit. Indeed, I moderately often cherry pick commits for substantive changes so I can highlight parts of a piece of work.

I avoid Git to the extent possible so I don't know what it offers for cherry picking changes, but I'll bet there is something in its slew of cobbled together tools that allows identifying and selecting changes to commit.