So, I figured out how to do it again. And now here I am making notes in public, where the world can Google it if they want to, and so can I.
I SSH'd into Server1, where I found the files I wanted stashed in this directory: /domains/t/h/thiswebsite.com/web/public_
Then I checked Server2 to find out where I wanted to put the files. Ah yes, in /home/thiswebsite/public_html (different directory structure on Server2, just to make my life more exciting apparently).
So I did this:
scp -rp /domains/t/h/thiswebsite.com/web/public_
Only then I found that user account, on domainname.com did not have shell access. In a moment's foolishness, I switched to root access. But this doesn't work (fortunately) - you can't just scp right in when you are root. The thing to do was to change email@example.com so that he had shell access. So I did that, and it worked. Hurray! A pile of monster files copied direct.
more helpful instructions here.