  Much remains to be understood about the nature of exoplanets smaller than
Neptune, most of which have been discovered in compact multi-planet systems.
With its inner ultra-short period planet b aligned with the star and two larger
outer planets d-c on polar orbits, the multi-planet system HD 3167 features a
peculiar architecture and offers the possibility to investigate both dynamical
and atmospheric evolution processes. To this purpose we combined multiple
datasets of transit photometry and radial velocimetry (RV) to revise the
properties of the system and inform models of its planets. This effort was
spearheaded by CHEOPS observations of HD 3167b, which appear inconsistent with
a purely rocky composition despite its extreme irradiation. Overall the
precision on the planetary orbital periods are improved by an order of
magnitude, and the uncertainties on the densities of the transiting planets b
and c are decreased by a factor of 3. Internal structure and atmospheric
simulations draw a contrasting picture between HD 3167d, likely a rocky
super-Earth that lost its atmosphere through photo-evaporation, and HD 3167c, a
mini-Neptune that kept a substantial primordial gaseous envelope. We detect a
fourth, more massive planet on a larger orbit, likely coplanar with HD 3167d-c.
Dynamical simulations indeed show that the outer planetary system d-c-e was
tilted, as a whole, early in the system history, when HD 3167b was still
dominated by the star influence and maintained its aligned orbit. RV data and
direct imaging rule out that the companion that could be responsible for the
present-day architecture is still bound to the HD\,3167 system. Similar global
studies of multi-planet systems will tell how many share the peculiar
properties of the HD3167 system, which remains a target of choice for follow-up
observations and simulations.
