The Turnitin API has a hissy fit if you try to create an assignment with a start date in the past - so the integration code won't allow creation with a date in the past. AFAIK they are improving this with their new api so the new version of the plugin that Turnitin are developing may not have this issue.
The due date is usually a bit more manageable - the code checks to make sure the date is greater than the time the assignment was created via cron + 10min and if not extends the due date to 1yr from now (probably a bigger extension than is needed!) If you had set a later due date by a couple of hours or days it would probably have used that date.
the code that manages dates is slightly better in the latest code in my git repo compared to the zip that is available in the plugins db.