An orthogonal integrated cleaving apparatus and methods of controlling such an apparatus are taught. The cleaving apparatus includes two cleaving devices mounted at right angles with respect to a mount facing a substrate to be processed. The non-metallic and/or brittle substrate is separated along two orthogonal axes without rotating or moving the substrate to a second machine by moving the mount or the substrate along the axes. Each cleaving device laser sequentially heats a surface of the substrate along a respective cutting axis, cools the cut area and then laser re-heats the cut area to form a clean break.