  The $B^0 \to K^{*0} \overline{K}^{*0}$ and $B^0_s \to K^{*0}
\overline{K}^{*0}$ decays are studied using proton-proton collision data
corresponding to an integrated luminosity of 3fb$^{-1}$. An untagged and
time-integrated amplitude analysis of $B^0_{(s)} \to (K^+\pi^-)(K^-\pi^+) $
decays in two-body invariant mass regions of 150 MeV$/c^2$ around the $K^{*0}$
mass is performed. A stronger longitudinal polarisation fraction in the ${B^0
\to K^{*0} \overline{K}^{*0}}$ decay, ${f_L = 0.724 \pm 0.051 \,({\rm stat})
\pm 0.016 \,({\rm syst})}$, is observed as compared to ${f_L = 0.240 \pm 0.031
\,({\rm stat}) \pm 0.025 \,({\rm syst})}$ in the ${B^0_s\to K^{*0}
\overline{K}^{*0}}$ decay. The ratio of branching fractions of the two decays
is measured and used to determine $\mathcal{B}(B^0 \to K^{*0}
\overline{K}^{*0}) = (8.0 \pm 0.9 \,({\rm stat}) \pm 0.4 \,({\rm syst})) \times
10^{-7}$.
