I found the answer.
It is an innocuous little setting in the email program on the WM 6.5 phone. First, you must have the 'Delete items off server' setting that Fwaits mentioned in post #3. But that won't work alone.
Open the Email program but don't choose an email account to view. Select Menu/Options. Now instead of choosing a specific account for editing, click the 'Storage' tab at the bottom of the screen for settings that control all accounts. Open the drop-down box under the item called 'Empty deleted items' and choose 'Immediately'. (I had it on 'Manually.) Now your IMAP messages will be deleted from your Gmail inbox when you delete them on your Windows Phone.