  In an accreting X-ray pulsar, a neutron star accretes matter from a stellar
companion through an accretion disk. The high magnetic field of the rotating
neutron star disrupts the inner edge of the disc, funneling the gas to flow
onto the magnetic poles on its surface. Hercules X-1 is in many ways the
prototypical X-ray pulsar; it shows persistent X-ray emission and it resides
with its companion HZ Her, a two-solar-mass star, at about 7~kpc from Earth.
Its emission varies on three distinct timescales: the neutron star rotates
every 1.2~seconds, it is eclipsed by its companion each 1.7~days, and the
system exhibits a superorbital period of 35~days which has remained remarkably
stable since its discovery. Several lines of evidence point to the source of
this variation as the precession of the accretion disc, the precession of the
neutron star or both. Despite the many hints over the past fifty years, the
precession of the neutron star itself has yet not been confirmed or refuted. We
here present X-ray polarization measurements with the Imaging X-ray Polarimetry
Explorer (IXPE) which probe the spin geometry of the neutron star. These
observations provide direct evidence that the 35-day-period is set by the free
precession of the neutron star crust, which has the important implication that
its crust is somewhat asymmetric fractionally by a few parts per ten million.
Furthermore, we find indications that the basic spin geometry of the neutron
star is altered by torques on timescale of a few hundred days.
