I feel confused. Consider the statement.
Matrices are normal iff they are similar to diagonal via multiplication by unitary matrices.
Any projection has a full set of linearly independent eigenvectors, which we can turn into orthonormal set via Gram-Schmidt process. Therefore, we can express a projection as a product of orthogonal (which are, of course, unitary) matrices and a diagonal one. Nevertheless, it need not be a normal matrix. For me, it looks like a contradiction to the result in quoting marks above.