This is not true for eXch.
eXch.cx does not keep logs, btw they already publicly proven that they are not cooperating with LE, and the user has the option of deleting all their data at the end of the order.
OK, good to know.
So I guess that when order data is deleted, there is nothing to
give to them in the first place.
I still do not recommend using this method for
anonymity purposes however, unless you are doing BTC <=> XMR. Because of the inherent risks of trusting a CEX (I mean, it's not trustless - you still have to take them by their word).