I'm just going to guess that Zodiark is one of the Twelve, and the Twelve should be Thirteen.
Why? That little comment Nabriales made about Lahabrea and Elidibus being "of this world". That implies, as a whole, Ascians are not of this world.
What does that have to do with the Twelve? Well, they famously left Hydaelyn. If, prior to the First Umbral Era, the Twelve (and presumably other Gods, who may well be Eikons at this point) lived alongside man in peace, until man started fighting, probably out of zealous devotion to a single God/Eikon, resulting in the Twelve leaving, since they'd recognize that they're (partly) the cause of the fighting... Well, that certainly seems how things should be, up until man ruined everything that is... The Ascians wish to return the world to how it was, so either way I think they have some connection to a time before the First Umbral Era... Zodiark being the forgotten member of the Twelve, who decided to return to see things put back to how they were? Fits nicely IMO, it works as a Final Fantasy XII reference, without being a direct copy, since it effectively flips things around (Zodiark being the rebel rather than Ultima). The Twelve left because it was the right thing to do, but Zodiark didn't want to, perhaps it was even his devout that were causing most of the problems, so they were sent away with him... Perhaps they're now the Ascians...
Of course, we should probably take mythology with a grain of salt... The Twelve didn't create Midgardsormr after all, he's a magical space dragon after all... Unless they did, and that's why he came from the stars, because that's where the Twelve left to... Seriously Midgardsormr, can you please start answering some questions? I don't know what's more unnerving... A taxi driver who awkwardly talks sports with you, when you have no understanding of said sports, or a taxi driver who gives you the silent treatment... You're the later Midgardsormr... Are you a serial killer? That's what it feels like...
Oh, and obviously Louisoix attempted to summon the Twelve... But well, Ysayle attempted to summon Saint Shiva, and Tiamat attempted to summon Bahamut... We know summoning is pretty bogus at this point...
So yeah... Prior to the First Umbral Era, Eikons and man live happily together, until they didn't for some reason. Zodiark is an Eikon, The Twelve are Eikons (#GaiusWasRight!). They left and forced Zodiark away with them, but he wants to return, and is presumably not happy about the whole affair, hence Ascians. They gave the Allag means to seal away Eikons too... Almost like they had some kind of grudge... Now, someone is probably going to point out that the Ascians want to release all the Eikons, but they don't... Lahabrea claimed the Primals were necessary for their plan (though he doesn't sound to happy about having to use them), but there is a distinction between Primal and Eikon... Unukalhai, who lets face it is, if not an Ascian, is at least working closely with Elidibus, claims that having Eikons unleashed would be bad, whats more he even suggests that the current Extremes aren't the work of the Ascians... Should I trust the little blighter? Probably not, but why would they write that in otherwise? If he really does have some connection with the Ascians, if he is honest about how this shouldn't be part of the Ascians plan (perhaps the other Ascians are misguided in attempting such things, but this isn't Zodiarks will?), then... Zodiark doesn't want Eikons to be released... Again, it's almost like he has a grudge against them...
And just because I can't go a moment without wanting to remind everyone that Free Companies are meant to be able to summon "Primals" eventually... This would mean Eikons are good... This would mean Eikons could be summoned by us to stop the Ascians... Might there be an Eikon Ifrit somewhere out there, and the current Ifrit is just an impostor? I sure hope so... Granted, Tiamat also claims the Eikons shouldn't be released, but she's been duped by the Ascians once before... Then there are those stranger readings at Urths Fount during 2.55... Why were they stranger? Was that researcher just that green? I'd have figured they'd all be pretty used to seeing the damage Primals caused by now... Odin isn't a Primal though, and that's probably what that was alluding to... But what if those readings were unusual because they suggested his presence was good? Guess we'll never know what was strange about them... Thanks Ilberd...
Oh right, the Warring Triad are Eikons... Hmmm, well in Final Fantasy VI they turned themselves to stone, given how hyped up they're being I kind of hope that's the case here too... If they had such limitless power, how'd the Allag seal them? I find it more likely they were already sealed when the Allag found them, and the Allag merely attempted to tap into their power while keeping them sealed. At the dawn of the First Umbral Era, the Twelve left Hydaelyn, while the Warring Triad turned themselves to stone, I think that would fit nicely... Unukalhai, and thus Elidibus/Zodiark, then don't want them being released because they'd probably be on-par with the Twelve and Zodiark and likely to deal with Zodiark... We perhaps also don't want them getting loose, since I can't imagine an Eikon showdown would be good for us, even if unlike Primals, Eikons aren't bad for Hydaelyn... Though I find it more likely that a certain Garlean Legate is going to absorb them at some point anyway...