What is the link between the homography computed over 2 images and the homography computed over the same images upside down?
I think when looking at the formula for the homography matrix:
$$
 H_{ba} = R - \frac{tn^T}{d}
$$
where $R$ is the rotation matrix by which $b$ is rotated in relation to $a$; $t$ is the translation vector from $a$ to $b$; $n$ and $d$ are the normal vector of the plane and the distance to the plane respectively (see Homography-3D plane to plane equation).

The relation between the two matrices lies in the normal vector of the plane. So you need to get the normal vector of the plane (out of the homography matrix), and apply the rotation to it, and then compute the homography matrix using the formula above. For the correct decomposing of the homography matrix, you can look at these code samples and this paper.