As ever, dear readers, welcome.
In the commentary included in a reply to Charlotte and Denis Plimmer of 8 February, 1967, Tolkien says that “the passages that now move me most” from The Lord of the Rings are:
“…the end of the chapter Lothlorien (I 365-7), and the horns of the Rohirrim at cockcrow.” (Letters, 376)
For me, it’s that second passage, just as it is the depiction of the Rohirrim in general which I find the most satisfying parts of Jackson’s film adaptation of the books.
As he based his depiction upon his native New Zealand, Jackson was faced with a geographical difficulty, however: Rohan is, basically, a great, grassy plain,

probably something like this—

which was the sort of place upon which ancient peoples from our Middle-earth, like the Skythians,

lived and bred their herds. The problem is that there is no area like that on New Zealand’s two main islands, but, with some very intelligent and creative location and camera work (as well as a huge construction project), we are given the look of Rohan, even without those missing plains.

I’m afraid that I don’t find the armor of the Rohirrim quite so convincing–

as in this image of Theoden, who appears to be wearing an awkward combination of plate armor and lamellar of leather (lamellar consists of small, overlapping pieces of metal or leather and is a very old method of armoring, being used, for example, by New Kingdom Egyptians—see this useful excerpt from a Nova special entitled “Building Pharaoh’s Chariot”: https://www.youtube.com/watch?v=gFxkYmqIX1w and here’s the whole excellent documentary: https://www.youtube.com/watch?v=0Loti-WBK_k ). To me, it appears much more like something from Frank Frazetta

than the Rohirrim described by Tolkien:
“The styles of the Bayeux Tapestry…fit them well enough, if one remembers that the kind of tennis-nets [the] soldiers seem to have on are only a clumsy conventional sign for chain-mail of small rings.” (letter to Rhona Beare, 14 October, 1958, Letters, 281)
Here’s what JRRT was thinking of—

and here’s a modern reconstruction by Christopher Rothero—

(The skirts are split to make it easier to wear while riding a horse.) So this is probably more like what Tolkien imagined–

(Could that Frazetta illustration have suggested the mode of conveyance of Jadis, the White Witch, in the 2005 film of The Lion, the Witch, and the Wardrobe?)

But then we hear the horns at cockcrow and we see the Rohirrim line up to charge and even if the armor looks off, the mass of horsemen certainly don’t.

(And here’s the scene if you don’t have it handy: https://www.youtube.com/watch?v=EmTz7EAYLrs Although missing the cockcrow, it seems.)
And mass is what we see, as in this overhead shot—

or do we?
Looking more closely at this image, there’s more order here than would first appear, and we have a clue to this from JRRT via Eomer, and two glosses:
1. when Eomer wants to speak to Aragorn alone, he tells his lieutenant, Eothain, he tells him: ‘Tell the eored to assemble on the path…” (The Two Towers, Book Three, Chapter 2, “The Riders of Rohan)
2. the 1966 Index: “a troop of Riders of Rohan (Index, 1154)
3. a note by Tolkien himself: “a ‘full eored’ in battle order was reckoned to contain not less than 120 men (including the Captain)” (“In a note to Cirion and Eorl and the Friendship of Gondor and Rohan” quoted in Hammond and Scull, The Lord of the Rings: A Reader’s Companion, 369)
This suggests that the Rohirrim cavalry were divided into distinct units, like Roman cavalry alae, which could then number 16 or 24 turmae, a turma having about 32 troopers.
Taking it that the Rohirrim were formed into eored (one of JRRT’s many Rohirric borrowings from Old English, the word meaning, among other things, “cavalry troop”), look again at the image of the massed horsemen from the film.
First, notice that the riders are actually in ranks. Next, notice that there seem to be divisions between blocks of mounted men. Then notice that those men are then divided into two larger units, the one following the other, with a gap between. What we’re seeing here, then, isn’t just a mob (although, once crashing into orc ranks it will become much more like one), but, instead, a series of units, one group forming the initial attacking force, the second its reserves—and, if you see it this way, you can then imagine that there’s more discipline here, like the famous (and near-fatal) charge of the British Light Brigade at the battle of Balaclava in October, 1854, where the Brigade was in three lines, as you can see from the image, so that the second and third lines could support the first.

This is, as I said, an image from the Jackson film, but JRRT had, himself, once been a cavalryman, if only briefly, in King Edward’s Horse. (Carpenter, Tolkien, 65)

He would have been well aware of the kind of training and discipline which went into being a trooper, although, at the beginning of the 20th century, when he was in his unit, there was considerable argument among military men about the role of cavalry. Some believed that, with the advent of weapons like the machine gun,

cavalry, if it continued at all, should be used as mounted infantry,

employing their horses for riding to a fight, then dismounting for combat. Others—often older officers–still believed that there was a role for the traditional mounted charge—in the right circumstances.

But the power of modern weapons won out and, throughout most of the Great War on the Western Front, cavalry either spent time in the rear, waiting for their big chance, or were converted to infantry. (For a stylized view of what might happen to cavalry should they attack machine guns, see this clip from the 2011 film War Horse: https://www.youtube.com/watch?v=geXxBueyR_A )
The Rohirrim, whom Tolkien once called “heroic ‘Homeric’ horsement” (letter to Milton Waldman, “probably written late in 1951”, Letters, 159) come from a time and place where bows were the most powerful missile weapons, however, and their own weapons would be, like Tolkien’s Normans on the Bayeux Tapestry, lances (or spears) and swords.

If we begin with the idea that the Rohirrim were grouped into units, like Roman cavalry, might they have been drilled like Roman cavalry and even fought like them? There seems to be evidence in The Lord of the Rings to suggest something like that and, in the second part of this posting, I want to consider what we might learn from a little close reading.
And, speaking of reading, thanks for doing so, as ever.
Stay well,
Use your spear underhand to deal with fleeing orcs,

(This very well done image is by Darren Tan, whose other work, much of it fantasy/sf, can be found at: https://www.deviantart.com/wraithdt )
And remember that, as always, there’s
MTCIDC
O