This map is based on a rough plan by Tolkien reproduced in volume VIII of The History of Middle-earth. A similar, but rather less detailed, second map also appears in the same volume. The fact that these maps are not identical shows that Tolkien was experimenting with different layouts when he drew these plans, and so they should by no means be taken as absolutely precise.
One peculiarity of the original sketch map is that is marked with an unconventional compass rose, showing 'W' and 'E' in their usual positions, but with 'N' and 'S' reversed. It's hard to be sure what Tolkien intended by this (strictly speaking, this arrangement should be impossible, unless the plan was drawn as if looking up from underground, which hardly seems likely). More plausibly, Tolkien drew the original map as if the cleft lay westward of the tunnels, but then realised that it should be to the east, and reversed the directions on the compass rose to indicate this. The map shown here therefore rotates the original so that the cleft and the Under-way point eastward rather than westward, though other valid orientations might be imagined.
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