Battles of the late Third Age

[Numenorean]

Good to see you are back Ru, and with a brilliant analysis of this battle and the geography too.

The thing which initially struck me about this clash was that despite the festering schemes and subterfuge of Saruman and Grima, the Rohirrim were still able to offer and put up a vigorous level of mobilisation when the need arose. Given the four years or so of dotage that Théoden endured prior to the attack, credit for this must lay primarily with Théodred and Éomer, though the other Captains of the Rohirrim that get a mention, Grimbold, Elfhelm and Erkenbrand also come across as being highly motivated, well trained and more than able to act upon their own initiative.

The pretext to Sarumans offensive is exactly as Rumil pointed out:

this was a prelude to the full-scale invasion of Rohan and was, apparently, fought with one aim in mind -
Saruman had given special orders that Theodred should at all costs be slain

To this end he deployed a special "company of men or orc-men...ferocious, mail-clad, and armed with axes." I wonder what race this company actually was? Tolkien describes them rather ambiguously as great orc-men in one breath then great axemen in the next, and he holds them distinct from the Uruks who charged down the eastern bank. At any rate these fiendish axe-wielders were successful in that they killed Théodred, but failed in their secondary objective of returning his body to Saruman as "Elfhelm himself...sprang up towards the knoll; and there he found Grimbold fighting two great axemen for possession of Théodreds body. One Elfhelm at once slew, and the other fell before Grimbold."
Then as Rumil states

Surprisingly, at a horn signal, the uruk-hai drew off and disappeared into the night. Likewise at the western end of the ford, Saruman's troops drew off. They had failed in their military objective to take the fords, but had succeeded in their political objective to kill Theodred. From Saruman's point of view this was a reasonable outcome

Reasonable for Saruman in the short term, but in the grand scheme of things it must surely go down as a disasterously missed opportunity for the Wizard. Rohan may well have fallen entirely had the Isengarders pressed on in full force instead of retreating under the assumption that killing the Kings Heir was enough to demoralise the proud Eorlingas.
For the Rohirrim however, I feel the opposite view holds for the outcome. In the short term they were badly beaten, and the loss of Théodred was great, yet if anything their resolve was hardened and their military structure remained intact as Erkenbrand immediately assumed command of the West-mark. All-in-all they were far from defeated, as they never actually withdrew from the west bank or the Fords of Isen, and more importantly still - they never lost hope.