The accreting X-ray pulsar Her X-1 shows two types of long-term variations, both with periods of ~35 days: 1) Turn-on cycles, a modulation of the flux, with a ten-day long Main-On and a five-day long Short-On, separated by two Off-states, and 2) a systematic variation in the shape of the 1.24s pulse profile. While there is general consensus that the flux modulation is due to variable shading of the X-ray emitting regions on the surface of the neutron star by the precessing accretion disk, the physical reason for the variation in the pulse profiles has remained controversial. Following the suggestion that free precession of the neutron star may be responsible for the variation in the pulse profiles, we developed a physical model of strong feedback interaction between the neutron star and the accretion disk in order to explain the seemingly identical values for the periods of the two types of variations, which were found to be in basic synchronization. In a deep analysis of pulse profiles observed by several different satellites over the last three decades we now find that the clock behind the pulse profile variations shows exactly the same erratic behavior as the turn-on clock, even on short time scales (a few 35d cycles), suggesting that there may in fact be only one 35d clock in the system. If this is true, it raises serious questions with respect to the idea of free precession of the neutron star, namely how the neutron star can change its precessional period every few years by up to 2.5% and how the feedback can be so strong, such that these changes can be transmitted to the accretion disk on rather short time scales.