It sounds very much like the web root for the web server has changed, and is pointing to an older version also on the HDD.
Which OS is this…? can you check the document root for the site in the web server software and confirm that it's pointing to the right folder…? can you use a system-level search to locate the 'other' copies of the php files which are seemingly on the server somewhere…?
Few random thoughts which might help…
Thanks for your input, Steve. The problem has been fixed.
In case anyone else has this same problem: It appears the culprit was Apache; apparently it was serving up pages from its cache in error. Some page versions it was serving were almost a year old! The client's IT ended up doing a restore of the whole system from a few days ago, and restoring the databases from today's end of the day backup. That seems to have cleared up the problem.