  We quantify, within mode coupling theory, how changes in the liquid structure
affect that of the glass. Apart from the known sensitivity to the structure
factor $S(q)$ at wavevectors around the first sharp diffraction peak $q_0$, we
find a strong (and inverted) response to structure at wavevectors \emph{below}
this peak: an increase in $S(q_0/2)$ {\em lowers} the degree of arrest over a
wide $q$-range. This strong sensitivity to `caged cage' packing effects, on
length scales of order 2d, is much weaker in attractive glasses where
short-range bonding dominates the steric caging effect.
