A razor includes a handle and detachable blade cartridge. The end of the handle has a pivot sphere upon which the blade cartridge is rotatably mounted, with freedom to pivot around three axes of the handle, and otherwise holds its orientation relative to the handle. The blade cartridge is held to the pivot sphere by a clevis having recesses defined in at least the inner side surfaces of its legs. The recesses, together with the clevis, are sized to admit the pivot sphere between them and to hold the sphere therein once admitted. The clevis and/or the pivot sphere can be formed of a resilient injection molded plastic. The range of motion of the blade cartridge can be limited by forming one or more stops on the pivot sphere to interact with the clevis and thereby interrupt its free rotation.