Recent studies have suggested that type Iax supernovae (SNe Iax) are likely to result from a weak deflagration explosion of a Chandrasekhar-mass white dwarf in a binary system with a helium (He)-star companion. Assuming that most SNe Iax are produced from this scenario, in this work we extend our previous work on the three-dimensional hydrodynamical simulation of ejecta-companion interaction by taking the orbital and spin velocities of the progenitor system into account. We then follow the post-impact evolution of a surviving He-star companion by using the one-dimensional stellar evolution code MESA. We aim to investigate the post-explosion rotation properties of a He-star companion in SNe Iax. We find that the He-star companion spins down after the impact due to the angular-momentum loss and expansion caused by the mass-stripping and shock heating during the interaction. This leads to the situation where the surface rotational speed of the surviving companion can drop to one-third of its pre-explosion value when it expands to a maximum radius a few years after the impact. Subsequently, the star shrinks and spins up again once the deposited energy is released. This spin-switching feature of the surviving He-star companions of SNe Iax may be useful for the identification of such objects in future observations.