Mail automatically deletes itself as soon as it enters into the Inbox.

Scenario:  A mailbox was recently created and every time mail is sent to this mailbox, it immediately deletes from the Inbox.

You have already verified that there are no mailbox rules, mobile devices, and applications that are causing this behavior.

Solution:  Check the RecipientType of the mailbox that was created by running the following command:

get-recipient iBroke

We had a mailbox that was incorrectly configured as a Room Mailbox and this mailbox needed to be a normal user mailbox. The automatic calendar processing was immediately deleting this message.  We resolved this issue by running:

Set-Mailbox iBroke -Type Regular