Square donuts and twistable holes

A mathematical donut is a rectangle of integral side length with a smaller rectangle (called the hole of the donut), also of integral side length, strictly inside it and with sides of the rectangles parallel to each other, where the area of the larger rectangle is twice that of the smaller. Necessary and sufficient conditions are determined for when the hole of the donut can be rotated $90^{\circ}$ and a donut still exists, and a complete classification of all square or square-holed donuts is given, with the square donut classification being intimately related to Pythagorean triples.