  New symmetries, norm computations and spectral information are obtained for
the Leray transform on a class of unbounded hypersurfaces in $\mathbb{C}^2$.
Emphasis is placed on certain distinguished measures, with results on operator
norm monotonicity established by proving new polygamma inequalities. Classical
techniques of Bernstein-Widder and Euler-Maclaurin play crucial roles in our
analysis. Underpinning this work is a projective geometric theory of duality,
which manifests here in the form of H\"older invariance.
