We investigate the mesogen reorientation behavior of thin films of liquid crystal elastomers (LCEs) under uniaxial elongation (z-axis) normal to their initial homeotropic alignment (x-axis). The stress-strain relation is characterized by three regions: (I) a small-strain regime exhibiting a linear relationship, (II) a moderate-strain region with a quasi-plateau stress, and (III) a large-strain region where the stress again increases. The Poisson's ratio mu(yz) is strain-dependent: mu(yz)approximate to 0.17 in region II, whereas in regions I and III, mu(yz) approximate to 0.5 in accordance with that of incompressible isotropic elastomers. The considerably small mu(yz) in region II is a consequence of the suppression of the transverse shrinkage in the y-direction (corresponding to the director rotation axis) during elongation. Such a deformation mode is considerably closer to pure shear than uniaxial stretching. The infrared dichroism measurement reveals that the mesogen primarily rotates within the x-z plane toward the z-direction in region II, whereas no further mesogen realignment takes place in region III. These results indicate that the director-rotation process is featured by a quasi-plateau stress and a pure shearlike deformation. The orientation order of the mesogens in the fully reoriented state is comparable to that in the unloaded state. The complete unloading, even after stretching up to region III, recovers the original homeotropic alignment.