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 a particular three-stage shape of compression curves, unusual for polycrystals with cubic structure. However, the role of twins remains a matter of debate and it is not clear if the same features appear in other testing conditions. We performed tensile tests on commercially-pure Ti samples cut along the rolling and the transverse direction, which yielded several unexpected results. In particular, the work hardening rate was found to be lower in the latter case, although the EBSD measurements revealed for them a larger volume fraction of twins. Also, the two kinds of specimens showed an opposite sign for the strain-rate effect on the proneness to the three-stage shape of the deformation curves. As a first approach, these observations are compared to the results derived from a simple Kocks-Mecking model. The possible role of twinning and dislocation glide on the anisotropy of mechanical behavior of titanium is then discussed.