Mail messages disappearing

Some addenda on this mystery:

First, if I do a search on a recipient or sender I know is on a message, I can see the messages in their respective boxes. However, if I close search on that name, the message disappears.

Second, if while I have it visible I mark it as Unread, it is of course bolded, and then it does not disappear; I can view it as it normally should be. Mark it as read, and it disappears. This occurs with only some messages in some mailboxes (I've been able to discover no rhyme, reason, or pattern for this, and it just began).

Third, I thought the problem might be the mailboxes, but I can move the messages to newly-created mailboxes and the problem moves with them. Also it infects new messages I move to that mailbox.

This is both baffling and really frustrating. Any suggestions welcome.

Thank you. I will give that a try this weekend, though I am somewhat skeptical that it is 3rd party app interference. Still, that's why the tests are worth running. I did get my Sent folder contents back last night, (apparently rebuild just took several hours for that), and that was the biggest scare, but other folders are still not showing contents I know they have (and which can be obtained by Searching on the senders or recipients). And Inbox still does not show messages contained in its POP and IMAP subfolders, once those messages have been read (not moved or deleted), which was the original symptom I have been trying to troubleshoot. I'll try a Safe Mode boot and see what happens.

Some progress. My initial problem was caused by filtering, turned on I don't know how. On the advice of an Apple tech, I turned it off, and my initial Inbox invisibility problem was solved. However, I still have mailboxes with invisible messages, messages I cannot see but can find by searching on senders and recipients, if I know them. Some I can recover by moving to the Inbox and putting in newly created mailboxes. No idea how to recover those in folders I don't have a rough idea of contents of, or why messages would be going invisible in the first place.

Problem solved. It was filtering throughout. I have no idea how it got turned on, or in particular how it got turned on for some mailboxes or not others, or became the default for new folders. Nor can I find much guidance on filtering (that is, apart from the use of rules) through Mail or otherwise online. But that was the problem, and disabling it mailbox by mailbox solves the problem, in case anybody else ever runs into the issue.

Yes, that's it. I found it has to be done for each affected folder; it's not a global operation. It would be really nice if there were discussion of this feature somewhere, but I've been unable to discover it.

Unfortunately, Mail behavior is exactly the same in Safe Mode as in regular mode. Doesn't appear to have done anything to solve the problem. I'd welcome more suggestions.