Abstract Directional non-reciprocity refers to the phenomenon where the motion in one direction differs from the motion in the opposite direction. This behavior is observed across various systems, such as one-way traffic and materials displaying electronic/optical directional dichroism, characterized by the symmetry of velocity vectors. Magnetic toroidal moments (MTMs), which typically arise from rotational spin arrangements, also possess the symmetry of velocity vectors, making them inherently directionally non-reciprocal. In this paper, we examine magnetic point groups that exhibit MTMs, subsequently leading to off-diagonal linear magnetoelectricity. Our focus is on the induction of MTMs through electric fields, magnetic fields, or shear stress, while enumerating the relevant magnetic point groups. The findings of our study will serve as valuable guidance for future investigations on directional non-reciprocity, MTMs, and off-diagonal linear magnetoelectric effects.