If reflected or flipped on the major axes and diagonals, these four polygons remain distinct. However, this is the same polygon each time. The last three are rotations of the first, with the rotation matrix built from the arctan of different pythagorean triple based fractions: -12/5, 63/16, and 4/3.
One way to determine if a lattice polygon has a different rotational embedding is to apply all of the arctan pythagorean fraction rotation matrices and to see if the points stay on the lattice. Is there an easier method?