Flipping Ice Cream Cake

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Advice For New Users

Here's an interesting puzzle given to me by John in the Azores. I'd been torturing him with several puzzles, so he retaliated with this:

One thing to be careful of is that two successive cut-n-flips of 40 degrees is not the same as a single cut-n-flip of 80 degrees. The flip doesn't just reverse top and bottom, it also reverses left and right.
If the angle you've chosen happens to divide exactly into 360 degrees then after one rotation you've turned the entire cake upside down, and on the second rotation you return each piece to its original position.

But what if your chosen angle is, say, 50 degrees? Or perhaps 75 degrees? Does the cake ever return to its original state (with extra cuts!)? Or does it just get more and more complicated?

What does it look like?

Part of the Farrago of Fragments.
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