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Plenty during the first half, and there are still some after the Cerebus Syndrome sets in. Sometimes count as Mood Whiplash. Like the time Bright is forced to defend the Argama against oranges. Five minutes later, Saegusa's dead.
This little exchange:
Mashymre: I am Mashymre Cello! The great Haman's faithful soldier! State your identity!
Judau: Judau Ashta! (thinks) Currently skipping school!
And if this sounds familiar, it's worth noting that Spaceballs came out a year later than ZZ. So to paraphrase Governor Le Petomane, "You'll be able to sue them!"
Chara/Kyara's introduction where she acts like some kind of reporter with a portable recorder. Mashymre approached her (after getting hit by an MS head carried on a crane and dragged back a little) and, knowing Haman will probably be listening to the recording later, grabbed it to say some inspiring things to her. Her reaction:
Kyara: ...It's off, y'know.
Kyara subjecting Judau to Marshmallow Hell to get him to help her escape from the Argama. It seems to have turned into a fetish, as she still claimed to enjoy it even after getting Brainwashed and Crazy.
This little moment:
Mashymre: (turns on the comms screen) This is Mashymre! Ready to launch! (giant screen and face appears) WHOA!!! SCALE IT DOWN!
Yazan's Villain Decay moments. For being such a badass in Zeta, here he's reduced to a hobo living off scraps and being the Comic Relief.
Zeta-Zaku◊, 'nuff said... both the Zeon forces and the Argama crew think it looks like an abomination, an insult to both the Zeta and the Zaku II. Making matters worse, it can't even transform (Zaku head is shaped incorrectly for that), or use the 360 degree monitor (the Zaku head uses the old mono-eye camera system, which preceded the 360 monitor).
The time they had to go undercover by disguising themselves as girls. Judau gets turned away because he looks "ugly". Iino, on the other hand...
The first time any of the Shangri-la kids pilot a large mobile suit counts. Judau's first move when he finally managed to get into the cockpit of Z Gundam is to... look for the button to close the cockpit. His next move? Accidentally hit the booster and crash into a wall. He won his first fight on sheer pure dumb luck. Elle's first time? Forget walking, all she could do was move the suit's arms up and down like one of those Chinese knockoff toy robots. Beecha and Mondo actually had better luck, as they had paid attention to the things the real pilots do. Too bad the suit they're using was supposed to be under repair and couldn't even close its cockpit hatch... am I seeing a pattern here?
Glemmy getting jealous of Judau after getting some "intel" from the defecting Beecha and Mondo. When they meet later on, Glemmy was plainly acting hostile while Judau was completely clueless to the reason since he and Roux aren't in that kind of relationship...yet.
For all its faults, the Moon Moon arc does contain one legitimately funny moment. When the Light Tribe demands that the Argama dismantle its "giants", Judau simply de-combines the Double Zeta, banking on the Light Tribe not understanding technology. In order to sell the ruse, Elle and Astonaige begin melodramatically sobbing over the loss of their "guardian god"... at which point Bright, Roux, and Emily enter, wondering what the hell's going on. Elle whispers to Bright, who joins in the "mourning"; he does the same for Roux, and she does the same for Emily. And the Light Tribe completely buys it.