  The Leray transform and related boundary operators are studied for a class of
convex Reinhardt domains in $\mathbb C^2$. Our class is self-dual; it contains
some domains with less than $C^2$-smooth boundary and also some domains with
smooth boundary and degenerate Levi form. $L^2$-regularity is proved, and
essential spectra are computed with respect to a family of boundary measures
which includes surface measure. A duality principle is established providing
explicit unitary equivalence between operators on domains in our class and
operators on the corresponding polar domains. Many of these results are new
even for the classical case of smoothly bounded strongly convex Reinhardt
domains.
