  We study the birational geometry of some moduli spaces of abelian varieties
with extra structure: in particular, with a symmetric theta structure and an
odd theta characteristic. For a $(d_1,d_2)$-polarized abelian surface, we show
how the parities of the $d_i$ influence the relation between canonical level
structures and symmetric theta structures. For certain values of $d_1$ and
$d_2$, a theta characteristic is needed in order to define Theta-null maps. We
use these Theta-null maps and preceding work of other authors on the
representations of the Heisenberg group to study the birational geometry and
the Kodaira dimension of these moduli spaces.
