Real elements in the mapping class group of T2

We present a complete classification of elements in the mapping class group of the torus which have a representative that can be written as a product of two orientation reversing involutions. Our interest in such decompositions is motivated by features of the monodromy maps of real fibrations. We employ the property that the mapping class group of the torus is identifiable with SL(2,Z) as well as that the quotient group PSL(2,Z) is the symmetry group of the Farey tessellation of the Poincaré disk.