  We construct a microscopic spin-exchange Hamiltonian for the quasi-1D Ising
magnet CoNb$_2$O$_6$ that captures detailed and hitherto-unexplained aspects of
its dynamic spin structure factor. We perform a symmetry analysis that recalls
that an individual Ising chain in this material is buckled, with two sites in
each unit cell related by a glide symmetry. Combining this with numerical
simulations benchmarked against neutron scattering experiments, we argue that
the single-chain Hamiltonian contains a staggered spin-exchange term. We
further argue that the transverse-field-tuned quantum critical point in
CoNb$_2$O$_6$ corresponds to breaking this glide symmetry, rather than an
on-site Ising symmetry as previously believed. This gives a unified microscopic
explanation of the dispersion of confined states in the ordered phase and
`quasiparticle breakdown' in the polarized phase at high transverse field.
