  The recently discovered dipole-octupole pyrochlore magnet Ce$_2$Hf$_2$O$_7$
is a promising three-dimensional quantum spin liquid candidate which shows no
signs of ordering at low temperature. The low energy effective pseudospin-1/2
description in a magnetic field is characterized by the XYZ Hamiltonian and a
Zeeman term where the dipolar local z-component of the pseudospin couples to
the local z-component of the applied magnetic field, while the local x- and
y-components of the pseudospin remain decoupled as a consequence of their
octupolar character. Using effective parameters determined in V. Poree et al.,
arXiv:2305.08261 (2023), remarkable experimental features can be reproduced, as
for instance the specific heat and magnetization data as well as the continuum
of states seen in neutron scattering. Here we investigate the thermodynamic
response to magnetic fields applied along the global [110] direction using
specific heat measurements and fits using numerical methods, and solve the
corresponding magnetic structure using neutron diffraction. Specific heat data
in moderate fields are reproduced well, however, at high fields the agreement
is not satisfactory. We especially observe a two-step release of entropy, a
finding that demands a review of both theory and experiment. We address it
within the framework of three possible scenarios, including an analysis of the
crystal field Hamiltonian not restricted to the two-dimensional single-ion
doublet subspace. We conclusively rule out two of these scenarios and find
qualitative agreement with a simple model of field misalignment with respect to
the crystalline direction. We discuss the implications of our findings for
[111] applied fields and for future experiments on Ce$_2$Hf$_2$O$_7$ and its
sister compounds.
