Response songs appear at first to be a different type of doubling than covers. The Velvet Underground's "I'm Waiting for the Man" and the Pixies' "Here Comes Your Man" seem to have little in common other than the circumstances under which the later song was written. Still, like covers, the force of the response requires something extra from the listener, who, I suppose, misses the point, if the original is unknown. Moreover it reminds us of the importance of musicians as not only performers, but listeners.
Here's Camera Obscura responding to Lloyd Cole.