It is well known that any rectangle can be cut into a finite number of pieces and reassembled into a rectangle of same area (for example, the Montucla method). A rectilinear polygon is defined here: https://en.wikipedia.org/wiki/Rectilinear_polygon

Question: Can we dissect any rectangle to another rectangle of same area if only rectangles can be used as intermediate pieces. What if we are allowed to use only rectilinear polygons as intermediate pieces?

It is well known that, for any two rectangles of the same area, the first can be cut into a finite number of polygonal pieces and reassembled into the other (for example, by Montucla's dissection).

Questions: Can we do the same if only rectilinear polygons can be used as intermediate pieces? Or if only rectangles can be used as intermediate pieces?