  We obtain sharp ranges of $L^p$-boundedness for domains in a wide class of
Reinhardt domains representable as sub-level sets of monomials, by expressing
them as quotients of simpler domains. We prove a general transformation law
relating $L^p$-boundedness on a domain and its quotient by a finite group. The
range of $p$ for which the Bergman projection is $L^p$-bounded on our class of
Reinhardt domains is found to shrink as the complexity of the domain increases.
