  In this paper the ubiquity of the Igusa quartic $B \subset \mathbb P^4$ shows
up again, this time related to the Prym map $\mathfrak p : \mathcal R_6 \to
\mathcal A_5$. We introduce the moduli space $\mathcal X$ of those quartic
threefolds $X$ cutting twice a quadratic section of $B$. A general $X$ is
$30$-nodal and the intermediate Jacobian $J(X)$ of its natural
desingularization is a $5$-dimensional p.p. abelian variety. Let $\frak j:
\mathcal X \to \mathcal A_5$ be the period map sending $X$ to $J(X)$, in the
paper we study $\frak j$ and its relation to $\frak p$. As is well known the
degree of $\frak p$ is $27$ and its monodromy group endows any smooth fibre $F$
of $\frak p$ with the incidence configuration of $27$ lines of a cubic surface.
Then the same monodromy defines a map $ \mathfrak j': \mathcal D_6 \to \mathcal
A_5$ of degree $36$, with fibre the configuration of $36$ 'double-six' sets of
lines of a cubic surface. We prove that $\frak j = \frak j' \circ \phi$, where
$\phi: \mathcal X \to \mathcal D_6$ is birational. This provides a geometric
description of $\frak j'$. Finally we describe the relations between the
different moduli spaces considered and prove that some, including $\mathcal X$,
are rational.
