In a pure functional language, a doubly-linked list is not that interesting. The idea of a doubly linked list is to be able to grab a node and go in either direction, or to be able to splice into the middle of a list. In a pure functionaly language you probably are better off with one of these two data structures:
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A singly linked list with a pointer in the middle, from which you can go either left or right (a variant of Huet’s “Zipper”)
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A finger tree, which is a mind-blowing data structure invented by Ralf Hinze and Ross Paterson.
I’m a big fan of the zipper; it’s useful in a lot of situations.