Yes, cutting from the incenter (a new term to me -- the center of a circle inscribed in a triangle, found where the bisections of each of the triangles angles intersect) to each edge at 90* yields three interchangeable pieces, and swapping two of them gives a mirror image of the original.
And Bravo! for the idea to turn the box inside out. I ruled out the cake flipping idea by specifying that it is iced, but never thought of this idea!!
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- Friday Puzzler -- Cake and Box