Conformally mapping from a rectangle to rectangle can be done as follows:
$r_2 = sn^{-1}(sn(r_1,m_1),m_2)$
where $m_1,m_2$ depend on the the input and output rectangle, respectively. (In particular, it maps $[-K(m_1),K(m_1)] \times [-K(1-m_1),0]$ to $[-K(m_2),K(m_2)] \times [-K(1-m_2),0]$).
The problem with this map is that its not symmetrical though. For example, doing an inverse image transform, we go from
$\text{goes to }$
Note that fact that the although it has left right symmetry, it does not have top bottom symmetry.
My question, therefore, is does there exist a function which conformally maps an arbitrary rectangle to another arbitrary rectangle, such that it does preserves the symmetry of its input? Also, if you know the specific function, that is even better!
Note: For the purpose of symmetry, you may assume the input is not a square.
EDIT: It appears in particular a center preserving conformal map would do the trick. Perhaps we could "move" the center point of one to the center point of the other using a conformal automorphism?