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I think this whole Primal-Twelve connection is an unfounded theory spurred by Gaius' claim that the twelve and the Primals are the same.
He may be right about the eikon thing, but I think the Twelve are a different kind of gods. I think they are a polytheistic take of the Abrahamic god. They exist but will not (or cannot) enter the mortal realm like Primals, so they have to work in mysterious ways. You could take it further and say that the Twelve themselves are aspects of a single God; Hyadelyn.
In contrast. what we called The primals were independent gods separate from Hyadelyn and the Twelve, and therefore are capable of entering the mortal plane. But Zordiark appeared and made them his own. Those eikons became Primals, aspects of the Prime Darkness Zodiark, but retain their abilities, (such as the aforementioned assuming of psychical form as well as tempering).
Basically, Hyadelyn and the 12 are one and the same, while the Primals eventually became extensions of Zordiark's will.
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Whether the Twelve and the Primals are the same or not, the twelve can be summoned. Or at least the Ascians think so. Remember, Lahabrea tricked a group of Ala Mihgan youth into trying to summon Rhalgar. The fact that Louisoix only summoned the power of the Twelve instead of the Twelve themselves was a choice on his part.
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Yeah, it was proven in 1.0 that the Twelve do at least exist in some form, and that Louisoix mentioned that it was indeed possible to summon them like a Primal, but to do so would enact a terrible cost on the planet itself, due to the huge amount of aether that would needed to be consumed would threaten Hydaelyn's very existence. Hence he used a shortcut at the Battle of Carteneau, summoning a portion of their power and not the actual deities themselves, as his 'plan B' to reseal Bahamut.
Naturally this wasn't enough sadly, as the 'End of an Era' trailer showed. Hydaelyn Herself however is not part of the Twelve - she's essentially the consciousness of the planet itself, making her a goddess in a sense, but not in a anthropomorphic way. She instead takes the form of a giant crystal - the source of all life itself (and the Echo too for that matter).
In some respects, Hydaelyn (and Zodiark too for that matter), are beyond the pantheon of the Twelve and exist in a separate hierarchy, but what exactly the Twelve really are - whether they really are as they're believed to be by most Eorzeans (virtual copies of the Greco-Roman pantheon), the same as the Primals, or some other form, remains to be seen.