We extend previous studies of the physics of interstellar cloud collisions by beginning investigation of the role of magnetic fields through 2D magnetohydrodynamical ( MHD ) numerical simulations . In particular , we study head-on collisions between equal mass , mildly supersonic diffuse clouds similar to our previous study . Here we include a moderate magnetic field , corresponding to \beta = p _ { g } / p _ { b } = 4 , and two limiting field geometries , with the field lines parallel ( aligned ) and perpendicular ( transverse ) to the colliding cloud motion . We explore both adiabatic and radiative ( \eta = \tau _ { rad } / \tau _ { coll } \simeq 0.38 ) cases , and we simulate collisions between clouds evolved through prior motion in the intercloud medium . Then , in addition to the collision of evolved , identical clouds ( symmetric cases ) , we also study collisions of initially identical clouds but with different evolutionary ages ( asymmetric cases ) . Depending on their geometry , magnetic fields can significantly alter the outcome of the collisions compared to the hydrodynamic ( HD ) case . In the ( i ) aligned case , adiabatic collisions , like their HD counterparts , are very disruptive , independently of the cloud symmetry . However , when radiative processes are taken into account , partial coalescence takes place even in the asymmetric case , unlike the HD calculations . In the ( ii ) transverse case , the effects of the magnetic field are even more dramatic , with remarkable differences between unevolved and evolved clouds . Collisions between ( initially adjacent ) unevolved clouds are almost unaffected by magnetic fields . However , the interaction with the magnetized intercloud gas during the pre-collision evolution produces a region of very high magnetic energy in front of the cloud . In collisions between evolved clouds with transverse field geometry , this region acts like a “ bumper ” , preventing direct contact between the clouds , and eventually reverses their motion . The “ elasticity ” , defined as the ratio of the final to the initial kinetic energy of each cloud , is about 0.5-0.6 in the cases we considered . This behavior is found both in adiabatic and radiative cases .