When Shinji and Kaworu reach dogma in Eva-13 to pull the spears, Kaworu, who was expecting to find one spear of each type, utters:
2本とも形状が変化して そろっている
Both spears, their shape changed, and now match
(translation mine).
This line always bothered me: Kaworu expects one spear of each type, but they are both clearly Longinus (albeit in the "javelin" or "closed" configuration); why does he say that both changed shape?
It seems as if Kaworu isn't yet aware at this point that they are both Longinus spears; as if Longinus and Cassius changed into something else, when -in reality- only the Cassius one changed (into Longinus)... or did it?
The Solution to The Puzzle
It turns out that the spears become of Longinus type only when they're pulled out of Lilith and adopt the "open" (forked) configuration, not before!
This is apparently supported by the storyboards, which clearly state that both spears "turn into Longinus" after Eva-13 pulls them out, so before that, they were something else:
To recap, in NTE lore (unlike in NGE), only the two-pronged (open) form is truly "The Spear of Longinus", while the "javelin" (wound up) form is some intermediate/indeterminate state that is neither Longinus nor Cassius.
Now, the interesting part is Why did they become Longinus when Eva-13 pulled them out? The most likely explanation is that it is because Eva-13 is the Eva of Dispair, so when it takes hold of a spear, it automatically becomes the spear of despair, i.e. Longinus. This also explains why when Eva-01 reclaims one of the spears from Eva-13 during the battle in negative space, it immediately morphs into Cassius type: Eva-01 is the Eva of Hope, counterpart to Eva-13.
Somehow in Q, Kaworu was not aware of Eva-13's true nature, and he expected that they could pull the two spears in their last known state.
Of course, this reaises more questions:
1) What's the deal with Mark.06? Why was it able to wield the spear of Cassius?
2) Why was the spear on the Moon in the Cassius configuration (wrapped, being carried) before it was given to Mark.06?
3) Why was the spear impaling Lilith at Dogma in 1.0 through 2.0 in the Longinus state, and not in the "Neutral" state like the two stabbed after Third Impact?
4) Why were at least two of the four spears involved in Second Impact in the Longinus state?
to name a few...
EDIT: Perhaps the storyboard above is super old news (I had never seen it, but then again...I haven't looked into the storyboards really), and the above analysis is common knowledge already and only I wasn't aware, in which case... feel free to bash me