  We report a search for the decays $B^0\to K^{*0} \overline{K}{}^{*0}$ and
$B^0\to K^{*0} K^{*0}$. We also measure other charmless decay modes with
$K^+\pi^-K^-\pi^+$ and $K^{+}\pi^{-}K^{+}\pi^{-}$ final states. The results are
obtained from a data sample containing $657 \times 10^6$ $B \overline B$ pairs
collected with the Belle detector at the KEKB asymmetric-energy $e^+e^-$
collider. We set upper limits on the branching fractions for $B^0\to K^{*0}
\kstarbar$ and $B^0\to K^{*0} K^{*0}$ of $0.81 \times 10^{-6}$ and $0.20\times
10^{-6}$, respectively, at the 90% confidence level.
