  We discuss questions arising from the work of Schellekens. After introducing
the concept of complementary representations, we examine $Z_2$-orbifold
constructions in general, and propose a technique for identifying the orbifold
theory without knowledge of its explicit construction. This technique is then
generalised to twists of order 3, 5 and 7, and we proceed to apply our
considerations to the FKS constructions $H(\Lambda)$ ($\Lambda$ an even
self-dual lattice) and the reflection-twisted orbifold theories $\widetilde
H(\Lambda)$, which together remain the only $c=24$ theories which have so far
been proven to exist. We also make, in the course of our arguments, some
comments on the automorphism groups of the theories $H(\Lambda)$ and
$\widetilde H(\Lambda)$, and of meromorphic theories in general, introducing
the concept of deterministic theories.
