Orientation of rectangle on conic section

Consider a conic section. There are 2 rectangles such that all of the 8 vertices of the 2 rectangles lie on the conic section. Further assume that the 2 rectangles have different orientation (ie. a side of one rectangle is not parallel to any side of the other rectangle). What are all the possible conic section for that to be the case?

Intuitively, the only possibility is the circle. It appears that the problem could potentially be solved by write down all the conic section equation and solving some complicated set of equations, but I would rather not do that. I am hoping for a nice and element geometrical approach to the problem, or one that base on group theory regarding isometry in Euclidean plane/space.

Consider a conic section. There are 2 rectangles such that all of the 8 vertices of the 2 rectangles lie on the conic section. Further assume that the 2 rectangles have different orientation (ie. a side of one rectangle is not parallel to any side of the other rectangle). What are all the possible conic section for that to be the case?

Intuitively, the only possibility is the circle. It appears that the problem could potentially be solved by write down all the conic section equation and solving some complicated set of equations, but I would rather not do that. I am hoping for a nice and element geometrical approach to the problem, or one that base on group theory regarding isometry in Euclidean plane/space.