  In the hindered magnetic dipole transitions of heavy quarkonia, the
coupled-channel effects originating from the coupling of quarkonia to a pair of
heavy and anti-heavy mesons can play a dominant role. Here, we study the
hindered magnetic dipole transitions between two $P$-wave bottomonia, $\chi_b(n
P)$ and $h_b(n^\prime P)$, with $n\neq n^\prime$. In these processes the
coupled-channel effects are expected to lead to partial widths much larger than
the quark model predictions. We estimate these partial widths which, however,
are very sensitive to unknown coupling constants related to the vertices
$\chi_{b0}(nP)B\bar B$. A measurement of the hindered M1 transitions can shed
light on the coupled-channel dynamics in these transitions and hence on the
size of the coupling constants. We also suggest to check the coupled-channel
effects by comparing results from quenched and fully dynamical lattice QCD
calculations.
