The crystallography of twinning is based on the concepts of simple shear and obliquity introduced by Mugge, Mallard and Friedel at the turn of the last century, with tensor mathematics later developed by Bilby, Bevis and Crocker in the 1960s. We propose a synthesis of these works by writing the three transformations (distortion, orientation and correspondence) as matrices in dyadic product forms. We show that a "normal" Friedelian mode is implicitly assumed. We introduce another mode called "tilted" that explains, with the simple twin index q = 1, some twins that were previously oddly reported with q = 2. We also interpret the type II twins, which are usually presented as the conjugate twins of type I twins, as simple shears a rational reciprocal plane, exactly as the type I twins are simple shears a rational direct plane. Finally, we explain why the term "twin" for variants inherited from a phase transformation is not appropriate, and we call for a generalization of the crystallography of twinning by considering epitaxial distortions and iso-orientation shears.