Thanks a lot for your input, CL. Now I know for sure I'm not being conned. That's fine.
I'm seconded to a public middle school in Shanxi by a recruiter. My contract is with the recruiter, not with the school itself. That's why I was on my guard.
Recruiters like the one above are fine for those of us who can't get Z visas on account of being over 60. Of course, that will depend on the actual province where the job is.