The plasticity of hexagonal materials is strongly anisotropic and involves different microscopic mechanisms such as mechanical twinning and dislocation glide. Twins are often considered to be responsible for various peculiar features of plastic flow, such as compression-tension anisotropy, a high work hardening rate, or a particular three-stage shape of deformation curves observed not only for single crystals but also for polycrystals. However, the role of twins remains a matter of debate. Moreover, most of the experimental results have been obtained in compression conditions and it is not clear whether the same features appear in other testing conditions. In the present work, tensile tests were performed on commercially pure alpha-Ti samples cut along the rolling and the transverse direction, and yielded several unexpected results. In particular, the work hardening rate was found to be lower in the latter case, although the EBSD measurements revealed a larger twin volume fraction in such samples. Besides, the two kinds of specimens showed an opposite sign of the strain-rate effect on the tendency to the three-stage work hardening. The possible role of twinning, dislocation glide, and aging of dislocations on the anisotropy of mechanical behavior of titanium is discussed. A mechanism based on the consideration of the dislocation glide anisotropy is suggested.