  The forward-backward asymmetry ${\cal A}_{\rm FB}$ in $B\to K^*\ell^+\ell^-$
decay is a sensitive probe of New Physics. Previous studies have focused on the
sensitivity in the position of the zero. However, the short distance effective
couplings are in principle complex, as illustrated by $B\to \rho\ell^+\ell^-$
decay within the Standard Model. Allowing the effective couplings to be
complex, but keeping the $B\to K^*\gamma$ and $K^*\ell^+\ell^-$ rate
constraints, we find the landscape for ${\cal A}_{\rm FB}(B\to
K^*\ell^+\ell^-)$ to be far richer than from entertaining just sign flips,
which can be explored by future high statistics experiments.
