Because the functions of polar materials are governed primarily by their polarization response to external stimuli, the majority of studies have focused on controlling polar lattice distortions. In some perovskite oxides, polar distortions coexist with nonpolar tilts and rotations of oxygen octahedra. The interplay between nonpolar and polar instabilities appears to play a crucial role, raising the question of how to design materials by exploiting their coupling. Here, we introduce the concept of 'polarization twist', which offers enhanced control over piezoelectric responses in polar materials. Our experimental and theoretical studies provide direct evidence that a ferrielectric perovskite exhibits a large piezoelectric response because of extended polar distortion, accompanied by nonpolar octahedral rotations, as if twisted polarization relaxes under electric fields. The concept underlying the polarization twist opens new possibilities for developing alternative materials in bulk and thin-film forms.