You are correct: doubly linked lists are almost always mutable data structures. However, you might consider trying to create an immutable range tree implementation instead. Perhaps you could use the original mutable one for testing.
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I'm walking through a computational geometry book, implementing each data structure in F# and the example that I was working on is a range tree. The book depicts the bottom rung of the tree as essentially a doubly linked list. Mid-way through my insert function I realized that I could not think of a way to implement a doubly linked list without using mutable values in my nodes. Is this correct? I think I'm right, and some quick searching of haskell and ocaml code seems to confirm this, I just wanted to make sure.