Compass/magnetic directions in Middle-earth

Question

Maps in The Lord of the Rings and The Hobbit have North/South/East/West marked on them, and I am wondering if anyone can tell me what these directions mean and how they are determined in-universe.

The reason I'm wondering is that I don't recall anyone ever using a compass, or (since they lose all their gear an alarming number of times) either lamenting the loss of a compass or feeling glad that a compass was not lost, nor any other type of navigational instrument for that matter.
I don't think they ever check bearings against the stars, although I haven't read The Lord of the Rings in a long while.
Anyway it's daylight when Thorin & Co. make a point of turning northwest after leaving Beorn's house. So what does it mean to them to "turn northwest," and how do they tell when they've done it?

Answer 1

West was the direction of Sunset and Valinor and was the principle compass direction. One would face West and everything else would follow from that.

The main compass direction is West. This is the direction of Valinor, and Númenor, and has extreme spiritual and cosmological significance.

They were led then to seats beside Faramir: barrels covered with pelts and high enough above the benches of the Men for their convenience. Before they ate, Faramir and all his men turned and faced west in a moment of silence. Faramir signed to Frodo and Sam that they should do likewise.

‘So we always do,’ he said, as they sat down: ‘we look towards Númenor that was, and beyond to Elvenhome that is, and to that which is beyond Elvenhome and will ever be. Have you no such custom at meat?’
_{The Lord of the Rings - Book 4, Chapter 5 - "The Window on the West"}

The elvish words for West and East are also related to the motion of the Sun. Númen, "West", is associated with sunset, and Rómen, "East", is associated with sunrise.

[Númenor] is a construction from the Eldarin base √NDU 'below, down; descend'; Q. núme 'going down, occident'; númen 'the direction or region of the sunset' +nóre 'land' as an inhabited area. I have often used Westernesse as a translation.
_{September 1965 Letter to Dick Plotz - Letters of JRR Tolkien #276}

• andúnë   ‘sunset, west’ in Andúnië, to which corresponds in Sindarin annûn, cf. Annúminas, and Henneth Annûn ‘window of the sunset’ in Ithilien. The ancient root of these words, ndu, meaning ‘down, from on high’, appears also in Quenya númen ‘the way of the sunset, west’ and in Sindarin dûn ‘west’, cf. Dúnedain. Adûnaic adûn in Adûnakhôr, Anadûnê was a loan from Eldarin speech.
• rómen   ‘uprising, sunrise, east’ (Quenya) in Rómenna. The Sindarin words for ‘east’, rhûn (in Talath Rhúnen) and amrûn, were of the same origin.

_{The Silmarillion - Appendix "Elements in Quenya and Sindarin Names"}

With West as the main compass direction, North and South follow as Right and Left.

The names of the letters most widely known and used were 17 n, 33 hy, 25 r, 10 f: númen, hyarmen, rómen, formen=west, south, east, north (cf. Sindarin dûn or annûn, harad, rhûn or amrûn, forod). These letters commonly indicated the points W, S, E, N even in languages that used quite different terms. They were, in the West-lands, named in this order, beginning with and facing west; hyarmen and formen indeed meant left-hand region and right-hand region (the opposite to the arrangement in many Mannish languages).
_{The Lord of the Rings - Appendix E - "Writing and Spelling" - "Note"}

[Quenya:] eldar ataformaiti epetai i hyarma ú ten ulca símaryassen úsië, an cé mo quernë cendelë númenna, ve senya i hyarma tentanë Melcorello ar cé mo formenna tentanes Amanna

[English:] The Elves were ambidexters; consequently, the left hand was not to them evil in their imaginations. On the contrary. For if one turned the face westwards as was usual, the left hand pointed away from Melkor (in the North), and if northwards, it pointed towards Aman (the Blessed Land).
_{The Nature of Middle-earth, Part 2, Chapter 3, "Eldarin Hands, Fingers and Numerals" (and also Vinyar Tengwar No. 49.)}

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In a somewhat earlier incarnation, in the very first surviving draft of The Hobbit (c.1929), we have a drawing of a compass rose with pictures for each of the points.

On this compass, it seems that:

• North points towards the Big Dipper
• South points to the sun
• West points to the Pelóri, the Mountains of Valinor (see this painting of Mount Taniquetil, made just one year earlier)
• East points to the Gates of Morn, where the sun emerges from every morning, (described in the Book of Lost Tales as "a great arch ... all of shining gold and barred with silver gates")

This is the identification agreed upon by most Tolkien Scholars, including John Rateliff, Douglas Anderson, Christina Scull, and Wayne Hammond. However it should be noted that Christopher Tolkien has a different interpretation for the East and West pictures. (More about that here.)

Answer 2

On a real planet, the directions (north, south, east, and west) are not defined by the planet's magnetic field. They are based around the rotation axis, which is the north-south axis of the planet. In the northern hemisphere,* the sun rises in the southeast and sets in the southwest. How far south it is depends on the time of year; it is farthest north at the summer solstice and farthest south at midwinter. The perpendicular bisector of the sun's daily arc across the sky runs due north-south, parallel to the rotation axis. This forms part of the basis for navigation by sextant, rather than magnetic compass. For rough alignment, precise tools like a sextant or astrolabe are not necessary; a skilled outdoor traveler can estimate directions fairly accurately from a couple brief observations of the sun and keeping track of the season, time of day, and approximate latitude.

Middle-earth exists on a round planet (or has at least since the seas were bent in the Second Age). However, it does not orbit the sun, but rather the sun is steered around the planet. However, there is no reason to believe that the sun does not follow the same kinds of arcs across Middle-earth that the real sun follows over the real world. So from the position of the sun, it should be straightforward to determine which direction is north, and the other directions follow directly from there (east-west just lying at right angles to north-south).

*Technically, this is only true year-round in the temperate zone between the tropics and the arctic.