Implications of heavy-quark spin-symmetry are investigated for the isoscalar charmonium-like state $X(3872)$ and the isovector bottomonium-like states $Z_b(10610)$ and $Z_b(10650)$ under the assumption of the latter being bound states of the nearby heavy meson-antimeson pairs. We formulate and solve a system of the integral equations for a coupled-channel problem involving the $P\bar P$, $P\bar V$, and $V\bar V$ channels (with $P$ and $V$ being either $D$ and $D^*$ or $B$ and $B^*$ mesons) to determine the scattering amplitudes in the channels with the quantum numbers $J^{PC} = 1^{++}, 1^{+-}, 0^{++}$, and $2^{++}$. The coupled-channel potentials incorporate the contact and one-pion exchange interactions derived in a chiral effective field theory approach and iterated to all orders. Once two contact terms at leading order are adjusted to reproduce the binding energies of the states used as input, the approach can be employed to predict the mass and the prominent contributions to
the width of the spin-partner states with the quantum numbers $J^{++}$ ($J=0,1,2$).