High mobility or the ease of movement of twin boundary in magnetic field, necessary for magnetically induced reorientation (MIR) is studied. In modulated 10M martensite two types of a-c twin boundaries, Type I and Type II are mobile in magnetic field. Here, we focus on the behavior of a single twin boundary in magnetic field at different temperatures. Using Type II, twin boundary MIR resulting in 6% field-induced deformation can occur in the field ~10 kA/m. For Type I, in contrast, the field more than ten times larger is needed. The effect of demagnetization is evaluated. In addition, the temperature dependence of the field needed to initiate twin motion is flat for Type II compared with sharp increase for Type I. The explanations for this sharply differing behavior are proposed.