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Scramble: L B2 L D2 B2 L' B2 R' B2 D2 R2 D' R U' B' L' D' R B L

x2 U R2 U F2 y2 U R’ F R // Cross
U R U’ R’ // F2L 1
y’ R U R’ U R U’ R’ // F2L 2
y’ U D’ L’ U L D // F2L 3
y’ U (missed this move in the reconstruction) R U R’ U2 R U R’ // F2L 4
y’ r U R’ U’ M U R U’ R’ // OLL
y F R U’ R’ U’ R U R’ F’ R U R’ U’ R’ F R F’ U2 // PLL

Spoiler: Solve 2 - DNF

Scramble: L U' B2 D2 U' L2 D' L2 B2 U F2 R2 F U L R2 B' L' U2 L2

I started the reconstruction while I was solving. I'm kind of stupid.

Scramble: L B2 L D2 B2 L' B2 R' B2 D2 R2 D' R U' B' L' D' R B L

x2 U R2 U F2 y2 U R’ F R // Cross
U R U’ R’ // F2L 1
y’ R U R’ U R U’ R’ // F2L 2
y’ U D’ L’ U L D // F2L 3
y’ U (missed this move in the reconstruction) R U R’ U2 R U R’ // F2L 4
y’ r U R’ U’ M U R U’ R’ // OLL
y F R U’ R’ U’ R U R’ F’ R U R’ U’ R’ F R F’ U2 // PLL

Spoiler: Solve 2 - DNF

Scramble: L U' B2 D2 U' L2 D' L2 B2 U F2 R2 F U L R2 B' L' U2 L2

I started the reconstruction while I was solving. I'm kind of stupid.

I checked it out: My 2 domino cubes don't do that move, but (as I knew before) my (Dian Sheng) Crazy 2x2x3 does that half-turn move. On that cuboid I considered it always as a legal turn - like mixup moves on a Mixup Cube.

But on a regular 2x2x3 I would not have considered these turns as legal, because it's obvious that they are not following the outer geometry of that puzzle. They are only possible because the cubies have not cube form on the inside.
Now I see that this counts for the Crazy 2x2x3 as well. These moves have nothing to do with the circle structure of that puzzle.

The illegal turns discussion makes me think of the Helicopter Cube (and related puzzles like the Curvy Copter, edge-turners of other shapes).

The cuts line up when you make an edge turn of about 71 degrees, allowing you to make a jumbling move on an adjacent axis. Was this intended by the two people (Adam and Katsuhiko) who first designed a puzzle with such a mechanism? It's uncertain whether Adam knew about it when he first announced the Helicopter Cube, but at least Katsuhiko already knew by then. They did consider jumbling moves to be valid and legal, and today it's standard practice to assume so as well.

The external appearance of cuts might not be a strong argument for whether moves should be allowed either. After all, there are puzzles like the Pandora Cube, where the appearance would suggest that it can't turn at all, and yet it's just a 3×3×3 shape mod.

But as a purist, I'd say that shapeshifting turns on a domino cube shouldn't be considered legal, even if you could execute the moves without running into physical obstructions. Besides purism, there's also the obvious argument that not every domino cube can do those moves. And yet we can't just impose a blanket ban on the domino cubes that can do M/S moves, because the most popular ones would then be banned! While M' U2 M doesn't let you go into puzzle states that aren't solvable with normal moves, it does provide a significant shortcut to the "proper" domino solution U2 R2 F2 R2 U2 R2 F2 R2 U2; allowing this would seem like a huge unfair advantage for those with M/S-capable hardware.

(Briefly going into a tangent on the copter cubes again, I've seen comments to the effect that the Curvy Copter Plus or whatever is the "true" unbandaged version of the Curvy Copter, but this is really a matter of drawing the line somewhere. You could, in theory, just keep adding new cuts to unbandage it, until the puzzle becomes unmeasurable dust at ω iterations, but clearly the most principled (canonical!) choice is to not add any cuts at all: which is to say, the original Curvy Copter!)