La neuralgia se me alborota.

[mwtzzz]

I know what it means, but I'd like an analysis of the grammar:

"La neuralgia se me alborota."

[AngelicaDeAlquezar]

What is it you don't understand, the pronouns?

"Se" is a pronominal particle which talks about something that I can't control. (You may find useful this discussion on the uses of "se".)

"Me" talks about the action of the "neuralgia" on me.

[mwtzzz]

I'd like to know why "La neuralgia se me alborota" but not "La neuralgia me alborota". ?

[AngelicaDeAlquezar]

Because it's "la neuralgia" who is acting weird on me, not the one making me act weird. That's what the "se" implies: I have no control on how it affects me.

La neuralgia se me alborota. -> The pain pops out from nowhere and I suffer.
My neuralgia starts acting up.

La neuralgia me alborota. -> The pain makes me nervous/anxious/restless...
My neuralgia shakes me up.

[mwtzzz]

Ah. Got it. The neuralgia is acting up (on me, for me), versus the neuralgia is making me act up.

But, both are valid, right?

In the first case, if you want to say "her neuralgia is acting up" you would write:
"La neuralgia se le alborota"
And if you want to say "their neuralgia is acting up" you would write:
"La neuralgia se les alborota"

And if you want to make a statement about neuralgia in general, is it okay to say:
"La neuralgia se alborota"

[AngelicaDeAlquezar]

Yes, it all depends on what you want to say. It's hard to have a generalized neuralgia or one that would be autonomic enough as to decide to act up by itself, but yes, all your sentences are grammatically correct.