The spontaneous formation of twins in La(2/3)Ca(1/3)MnO3 films below T(S) approximately 105 K is observed by a magneto-optical technique. The twins are revealed as stripes along the {110} directions where magnetization tilts out of the film plane due to the stresses in twins. Their appearance is associated with a martensitic phase transition in the film triggered by the cubic-to-tetragonal transition in the SrTiO3 substrate. It is found that magnetization of the films proceeds by inhomogeneous rotation of magnetic moments. This is due to the presence of microscopic structural inhomogeneities. Their dominating role in the low-temperature transport can explain small effects of the transition at T(S) on the resistivity.