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 ξ-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 ξ of the order of 10. Further increase in ξ 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 ξ 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.