We report new heat capacity measurements on single crystal Ce$_2$Zr$_2$O$_7$
down to $\sim$ 0.1 K in a magnetic field along the $[1,\bar{1}, 0]$ direction.
These new measurements show that the broad hump in the zero-field heat capacity
moves higher in temperature with increasing field strength and is split into
two humps by the $[1,\bar{1}, 0]$ field at $\sim$ 2 T. These separate features
are due to the decomposition of the pyrochlore lattice into effectively
decoupled chains for fields in this direction: one set of chains
($\alpha$-chains) is polarized by the field while the other ($\beta$-chains)
remains free. Our theoretical modelling suggests that the $\beta$-chains are
close to a critical state, with nearly-gapless excitations. We also report new
elastic and inelastic neutron scattering measurements on single crystal
Ce$_2$Zr$_2$O$_7$ in $[1, \bar{1}, 0]$ and $[0, 0, 1]$ magnetic fields at
temperatures down to 0.03 K. The elastic scattering behaves consistently with
the formation of independent chains for a $[1, \bar{1}, 0]$ field, while the
$[0, 0, 1]$ field produces a single field-induced magnetic Bragg peak at $(0,
2, 0)$ and equivalent wavevectors, indicating a polarized spin ice for fields
above $\sim$ 3 T. For both $[1, \bar{1}, 0]$ and $[0, 0, 1]$ fields, our
inelastic neutron scattering results show an approximately-dispersionless
continuum of scattering that increases in both energy and intensity with
increasing field strength. By modelling the complete set of experimental data
using numerical linked cluster and semiclassical molecular dynamics
calculations, we demonstrate the dominantly multipolar nature of the exchange
interactions in Ce$_2$Zr$_2$O$_7$ and the smallness of the parameter $\theta$
which controls the mixing between dipolar and octupolar degrees of freedom.
These results support previous estimates of the microscopic exchange
parameters.