Internally Architectured Materials (IAMs) that exhibit different friction forces for sliding in the opposite directions are proposed. This is achieved by translating deformation normal to the sliding plane into a tangential force in a manner that is akin to a toothbrush with inclined bristles. Friction asymmetry is attained by employing a layered material or a structure with parallel 'ribs' inclined to the direction of sliding. A theory of directionally asymmetric friction is presented, along with prototype IAMs designed, fabricated and tested. The friction anisotropy (the -coefficient) is characterised by the ratio of the friction forces for two opposite directions of sliding. It is further demonstrated that IAM can possess very high levels of friction anisotropy, with of the order of 10. Further increase in is attained by modifying the shape of the ribs to provide them with directionally dependent bending stiffness. Prototype IAMs produced by 3D printing exhibit truly giant friction asymmetry, with in excess of 20. A novel mechanical rectifier, which can convert oscillatory movement into unidirectional movement by virtue of directionally asymmetric friction, is proposed. Possible applications include locomotion in a constrained environment and energy harvesting from oscillatory noise and vibrations.