  We investigate the four-body decays $B_{(s)} \rightarrow (K\pi)_{S/P}
(K\pi)_{S/P}$ in the $K\pi$-pair invariant mass region around the $K^*(892)$
resonance in the perturbative QCD (PQCD) approach, where $(K\pi)_{S/P}$ denotes
a $S$- or $P$-wave $K\pi$ configuration. The contributions from the $P$-wave
resonance $K^*(892)$ and from possible $S$-wave components are included into
the parametrization of the $K\pi$ two-meson distribution amplitudes, which are
the nonperturbative inputs in this formalism. We calculate the branching ratio
for each resonance and observe sizable $S$-wave contributions to the $B_s$
modes, well consistent with recently available LHCb data. We also deduce the
polarization fractions for the $K^*\bar{K}^*$ final states, and compare our
predictions to those in previous PQCD analyses of the corresponding two-body
$B\to K^*\bar{K}^*$ decays. The polarization puzzle associated with the two
$U$-spin related channels $B^0\rightarrow K^{*0}\bar{K}^{*0}$ and
$B^0_s\rightarrow K^{*0}\bar{K}^{*0}$ still exists. In addition to direct $CP$
asymmetries, triple-product asymmetries and $S$-wave induced direct $CP$
asymmetries originating from the interference among various helicity
amplitudes, are presented. It is shown that true triple-product asymmetries are
rather small, while direct $CP$ asymmetries and $S$-wave-induced direct $CP$
asymmetries are significant in some decays. Our results will be subject to
stringent tests with precise data from $B$ factories in the near future.
