Zippers can be used to implement a two-way navigational list. The example in the article doesn’t support cycle, but it should be very easy to extend it by adding a clause which matches on the empty list in the second element.

The idea is to keep a tuple in the shape of {visited::[any], remaining::[any]}. So for example, in a tuple {[3, 2, 1], [4, 5, 6]} you’re currently at the element 4, while previously visited 1, 2, 3 (in that order). The next operation will move 4 to the top of the first list, so you’ll get {[4, 3, 2, 1], [5, 6]}. If after the move the second list is empty, then you just reverse the first list, and return {[], reversed_first_list}. So next({[5, 4, 3, 2, 1}, [6]) will return {[], [1, 2, 3, 4, 5, 6]}, and you’re back at the start.

You can take the similar approach to move in the opposite direction. The prev operation pops the head off the first list and pushes it at the top of the second list, unless the first list is empty, in which case you need to keep the last element of the second list, reversing all others into the first list.

Supporting insert and delete is as easy as pushing/popping to/from the top of the second list.

Another very interesting structure are the Finger Trees, that allow for efficient implementations of deques (double ended queues), priority queues or lists with O(1) size information - http://www.staff.city.ac.uk/~ross/papers/FingerTree.html There are couple implementations in erlang in the wild.

It could be interesting to see some Elixir based implementations with all the protocols.

Also worth noting that this is very similar to the implementation of a queue in functional programming. For more information on the underpinnings of such structures I highly recommend: https://www.cs.cmu.edu/~rwh/theses/okasaki.pdf