Critical Issue: Folders delete themselves, files create themselves as folders

When uploading files (largely PDF files, but also some files when in ZIP format), the web-based file system interface creates folders named the same as the file without uploading the file itself. 

I can keep deleting the folder, but anytime I upload the same file (and some but not all others) I tend to get the same result. 

Any ideas?  I have not been able to establish a WebDav client connection to narrow the problem from any WebDav (Windows) client I have tried (perhaps a symptom)? 

**** Since I saw the above issue, I tried to rename one of the folders via the web interface.  I got a generic error, and now the folder and its child objects are gone.  All public, site-wide attachments and images dropped...

I can't seem to be able to create another folder to start over, as I am not offered the option (disabled).

I could try to go back to the backups if I knew exactly what to restore.