ICloud mail: iPhone creates Deleted Messages folder for Trash mail?

Question

I had a problem with one iCloud email account which I've spent half a day truing to sort so though I'd post what happened here in case anyone knows how it happened or in case anyone has the same problem.  The problem centres around the Trash folder.
On all the other accounts (all iCloud email) the folders are replicated and all work.  So Sent, Archive, Drafts and Trash are there.  Deleting a mail on the iPhone, on the Mac or the Cloud puts the mail in the Trash (with the bin icon) and it's visible on all three devices in the same location (albeit the phone uses the Deleted Messages name for trash which is a bit dumb IMO considering it's called Trash everywhere else!!).
One account though seems to be in a mess.  In the Cloud the folder structure is correct and all with the icons.  The first problem is when adding the account to the phone.
First up, when selecting the various behaviours you can see the Deleted Messages icon with the bin icon to select for deleted messages (I do think this is confusing though - why not use the same folder name as appears in IMAP mail - Trash???)  BUT there is also a folder called Trash that the phone can see in IMAP but that doesn't exist in the iCloud folder structure.  Why this folder is even visible I don't know because on iCloud mail there are no user created folders at all, just the default icons and folder names.
When I send a message to the address on the iPhone and delete it from there the iPhone creates a Deleted Messages folder that is visible on iCloud mail.  When I select the bin/trash icon on mail on the phone the mail is in there.  The bin / Trash folder on iCloud remains empty and the Deleted Messages folder has the deleted mail.
So the first issue seems to be that the phone is seeing a folder that isn't there or that somehow the correctly named Trash folder with the icon has become named Trash when the phone sees it, and has a generic folder icon.  Either way, I can't delete this Trash folder.
How it seems to have got fixed was that after trying all manner of things I went to Preferences on iCloud.  I told iCloud to store messages in Deleted Messages.  I then deleted the Deleted Messages folder then suddenly this Trash folder the phone had been seeing became visible.  I then deleted that too and on checking back the Preferences had defaulted back to Trash, the only available option now. 
Having gone back to the iPhone the bin icon is still next to the Deleted Messages folder but that Trash folder had gone.  There were no other user defined folders either
I then tested sending messages to the account and deleting them from the phone, Mac and Cloud and they all went to the right place (Trash) and were all visible.
I'd had a problem ages ago where random mailboxes were appearing under the separate email names BELOW where the default boxes are listed and below any On My Mac folders.  I'm wondering whether at some stage, with an old Google hangover, whether I'd selected 'Use this Mailbox for' when these random boxes appeared which might have renamed or messed with the routing of boxes in the Cloud.

Answer 1

That's the way it works. Push notifications are only triggered when you get "new" mail. Other actions, like deleting, reading, moving to a folder do not trigger a Push notification. Those actions only change things when you open the mail app.