The class of memoryless heuristics for maintaining a Doubly-Linked List in an approximately optimal order is studied. Two mappings that relate Singly-Linked List and Doubly-Linked List heuristics are defined, and theorems involving these mappings are presented. A new heuristic referred to as the Swap heuristic for the Doubly-Linked List is introduced, and its asymptotic distribution has been derived.