  We present evidence for the existence of two ~35 day clocks in the Her X-1/HZ
Her binary system. ~35 day modulations are observed 1) in the Turn-On cycles
with two on- and two off-states, and 2) in the changing shape of the pulse
profiles which re-appears regularly. The two ways of counting the 35 day cycles
are generally in synchronization. This synchronization did apparently break
down temporarily during the long Anomalous Low (AL3) which Her X-1 experienced
in 1999/2000, in the sense that there must have been one extra Turn-On cycle.
Our working hypothesis is that there are two clocks in the system, both with a
period of about ~35 days: precession of the accretion disk (the less stable
"Turn-On clock") and free precession of the neutron star (the more stable
"Pulse profile clock"). We suggest that free precession of the neutron star is
the master clock, and that the precession of the accretion disk is basically
synchronized to that of the neutron star through a feed-back mechanism in the
binary system. However, the Turn-On clock can slip against its master when the
accretion disk has a very low inclination, as is observed to be the case during
AL3. We take the apparent correlation between the histories of the Turn-Ons, of
the Anomalous Lows and of the pulse period evolution, with a 5 yr
quasi-periodicity, as evidence for strong physical interaction and feed-back
between the major components in the system. We speculate that the 5 yr (10 yr)
period is either due to a corresponding activity cycle of HZ Her or a natural
ringing period of the physical system of coupled components. The question
whether free precession really exists in neutron stars is of great importance
for the understanding of matter with supra-nuclear density.
