  We describe the geometry of all type II common sector backgrounds with two
supersymmetries. In particular, we determine the spacetime geometry of those
supersymmetric backgrounds for which each copy of the Killing spinor equations
admits a Killing spinor. The stability subgroups of both Killing spinors are
$Spin(7)\ltimes \bR^8$, $SU(4)\ltimes \bR^8$ and $G_2$ for IIB backgrounds, and
$Spin(7)$, SU(4) and $G_2\ltimes \bR^8$ for IIA backgrounds. We show that the
spacetime of backgrounds with spinors that have stability subgroup $K\ltimes
\bR^8$ is a pp-wave propagating in an eight-dimensional manifold with a
$K$-structure. The spacetime of backgrounds with $K$-invariant Killing spinors
is a fibre bundle with fibre spanned by the orbits of two commuting null
Killing vector fields and base space an eight-dimensional manifold which admits
a $K$-structure. Type II T-duality interchanges the backgrounds with $K$- and
$K\ltimes\bR^8$-invariant Killing spinors. We show that the geometries of the
base space of the fibre bundle and the corresponding space in which the pp-wave
propagates are the same. The conformal symmetry of the world-sheet action of
type II strings propagating in these N=2 backgrounds can always be fixed in the
light-cone gauge.
