pop/imap gmx utter confusion

Hello

In short:

Can't set up a gmx imap account, yet pop account retrieves mails from imap account, but doesn't sync other folders with gmx.

In long:

1 / After syncing my mew iPhone 4 with my mac for the first time, 3 of 4 imap accounts are set up correctly ont he iPhone. The 4th however—the gmx.de one—is turned into pop and doesn't work at all. Since I couldn't find any option to turn it into an imap account, I

2/ deleted it and set it up anew, on the iPhone itself. In that process, the iPhone asks for Name, Address, Password, Description.

2a / First time I did this, result was the same as before. I get an error message that says that it can't connect.

2b / Second time, I put the imap server address (imap.gmx.net) into the Description (which in fact I thought to be a random name just for me to recognize, but one never knows … ): and really, now it started retrieving mails from gmx despite the account settings still tell it's a pop account. It does retrieve and (with some delay) send mail, but it doesn't sync folders (mails sent from the computer don't show, other than with the three imap accounts). (I must add though, that later changing that description (imap.gmx.net) into some random letters didn't change anything: it still got messages. 2c / And btw., once I changed the host name in the incoming mail server settings from pop.gmx.net into imap.gmx.net, iPhone gives an error message. But other than explained here (http://forum.gmx.com/forum/posts/list/3638.page), tapping ok here does't open up a choice between imap and pop.

3 / To push the confusion to a next level, I turned this half capable mail account off and set up another new one, with the same settings as in step 2a. It behaves like 2b.

I don't know what to do, and I can't find anything on line.

Many thanks in advance.