"But Legolas stood beside him, shading his bright elven-eyes with his long slender hand, and he saw not a shadow, nor a blur, but the small figures of horsemen, many horsemen, and the glint of morning on the tips of their spears was like the twinkle of minute stars beyond the edge of mortal sight." ( The Two Towers)

All of the offensive and defensive weaponry of the people of Rohan is geared towards fighting from horseback, from the round shield, designed to avoid hurting the horse with sharp edges, to the short light bow.

However, as it was for the Anglo-Saxons, who inspired JRR Tolkien in his creation of this people, the primary weapon of the Rohirrim is the spear, made of ash, the most distinguishing feature of an éored of Rohirrim: the force of a Rohan unit is measured in "spears", not "men" or "horses". Spear pallets are also used in battle to carry the dead and wounded off the field. In the "Lord of the Rings" movies the spears are characterized by distinctive cutouts in the point, like the arrows, thus reducing precision and range but making them lighter to carry.

Finally, the sword is the element that mostly carries a symbolical and ritual significance. The Rohirrim sword are said in the book to be "long" like actual cavalry swords, to give more reach to the rider: Éomer often "leans" on his when on foot. Hilts are decorated with green gems.
In the movies, like most Rohan artefacts, the swords have been embellished with horse designs on pommel, guard, scabbard and buckles, all parts made of bronze. The blade has a diamond-shaped cross-section and is unmarked, with no evidence of pattern-welding.
In this version the whole sword is short, about 3 feet long, two-edged, only slightly tapering, with a rounded point; a choice possibly motivated by the need of a more distinctive weapon compared to the long swords of other Men. The length is that of a Saxon single-edged seax, though the hilt and general appearance also recall longswords such as the one found at Sutton Hoo.


King Théoden's Sword

Háma knelt and presented to Théoden a long sword in a scabbard clasped with gold and set with green gems. 'Here, lord, is Herugrim, your ancient blade...' (The Two Towers)

A scene very similar to this in the movie "The Two Towers" is the focus of Théoden's return to his full faculties and kingliness. Herugrim (Old English "fierce sword") also helps creating a very moving and legendary moment just before the Ride of the Rohirrim, thanks to the inspired idea of actor Bernard Hill: "The king touching everybody's spear, it might be a Rohan tradition, that kind of thing. I was thinking in those kind of terms; that the king gives his spirit and sword to them, that he goes into their spirit somehow through the spear, and we're all in this together. This is it, we're all going to die, but you've got the king's spirit in you."

From a technical point of view, it might be worthy of notice that the actor is left-handed and Théoden's equipment is modified accordingly, sword hanging from his right side. The sword appears as a typical Rohan sword, simple but elegantly ornate, with a heavier guard crafted in the likeness of two horses' heads touching. Warren Mahy, designer-sculptor at WETA, points out that it is heavy but well-balanced, with the balance towards the guard.


Eomer's Sword

Out of doubt, out of dark to the day's rising
I came singing in the sun, sword unsheathing.
To hope's end I rode and to heart's breaking:
Now for wrath, now for ruin and a red nightfall!
These staves he spoke, yet he laughed as he said them. For once more lust of battle was on him; and he was still unscathed, and he was young, and he was king: the lord of a fell people. And lo! even as he laughed at despair he looked out again on the black ships, and he lifted up his sword to defy them.
And then wonder took him, and a great joy; and he cast his sword up in the sunlight and sang as he caught it... (The Return of the King)

Gúthwinë (Old English "battle friend") is the name of Éomer's sword, which he pledges to draw in battle side by side with Aragorn, and finally does at Helm's Deep. Its movie version can be easily distinguished from Théoden's and Éowyn's swords by its guard with two horse heads rearing on opposite sides on the blade. Warren Mahy describes it as a Viking-like short sword, one-handed, with no guard, easy to swing from horseback. Though maybe the most memorable image of movie-Éomer, the equivalent of the above-quoted scene, is of him hefting his spear in one single move and charging the Orcs on the Pelennor Fields, screaming and, I like to think, laughing.


Sources:
"The Lord of the Rings", by JRR Tolkien, Unwin Paperbacks, 1979.
"The LOTR Weapons and Warfare", by Chris Smith, Houghton Mifflin Company, Boston, 2003.
"The Anglo-Saxon Thegn AD 449-1066", by Mark Harrison, illustrated by Gerry Embleton, Osprey Warrior Series 5.
"The Lord of the Rings" Extended Version DVD, Disc 3 of "The Two Towers", Design Gallery
A conversation with Bernard Hill
The Encyclopedia of Arda
Lord of the Rings Costume Research Home
The Council of Elrond
Lisa's Video Frame Capture Library
The British Museum