As opposed to Wald’s cosmic no-hair theorem in general relativity, we find that the Horndeski theory admits anisotropic inflationary attractors if the Lagrangian depends cubically on the second derivatives of the scalar field. We name such a solution a self-anisotropizing inflationary universe, because anisotropic inflation can occur without introducing any anisotropic matter fields such as a vector field. Next, we investigate perturbations in homogeneous anisotropic background, and find that speed of gravitational waves depend on direction of its propagation. This investigation reveals, however, that the speed of gravitational waves in some directions diverges in the self-anisotropizing inflationary background and it suggests unfitness for representing our present universe. On the other hand, it is also revealed that as long as our universe is anisotropic, speed of gravitational waves can have angular dependence. Such a nature is not seen in general relativity, and it shows us abundance of phenomenology in Horndeski theory.