  We consider the relationship between the conjectured uniqueness of the
Moonshine Module, ${\cal V}^\natural$, and Monstrous Moonshine, the genus zero
property of the modular invariance group for each Monster group Thompson
series. We first discuss a family of possible $Z_n$ meromorphic orbifold
constructions of ${\cal V}^\natural$ based on automorphisms of the Leech
lattice compactified bosonic string. We reproduce the Thompson series for all
51 non-Fricke classes of the Monster group $M$ together with a new relationship
between the centralisers of these classes and 51 corresponding Conway group
centralisers (generalising a well-known relationship for 5 such classes).
Assuming that ${\cal V}^\natural$ is unique, we then consider meromorphic
orbifoldings of ${\cal V}^\natural$ and show that Monstrous Moonshine holds if
and only if the only meromorphic orbifoldings of ${\cal V}^\natural$ give
${\cal V}^\natural$ itself or the Leech theory. This constraint on the
meromorphic orbifoldings of ${\cal V}^\natural$ therefore relates Monstrous
Moonshine to the uniqueness of ${\cal V}^\natural$ in a new way.
