The aim in this particular kind of puzzle is to compose a piece of printed paper in order to obtain an image of the target. In principle, Rubik's Magic could belong to this category.
A better example is the 1-2-3-4 game. The operation to do is to fold a square piece of paper so that the four squares with numbers are near each other with no empty spaces and form a square. This puzzle is quite complex.
Another example of composition puzzle is what everyone probably has at home: folded prospectuses and city maps. Although the folding direction is often visible in already folded points, it could be extraordinarily difficult to fold the paper back into its original form.
Some examples are: