The Earth’s upper mantle is known to exhibit elastic anisotropy, which is common-ly attributed to the presence of Lattice Preferred Orientations (LPO). Such anisotropy is revealed in recordings of seismic waves that travel through the mantle with speeds that depend on propagation and/or polarization direction. The development of LPO is due to the plastic deformation of mantle minerals associated with large-scale convec-tive flow. Both olivine and pyroxene crystals exhibit an orthorhombic structure and have only a few slip systems available for dislocation creep. This leads to very high viscoplastic anisotropy at the grain scale, so that an upper mantle region with strong seismic anisotropy (i.e., pronounced LPO) may also exhibit a large effective viscoplas-tic anisotropy which may manifest itself as differences in effective viscosities of up to one or two orders of magnitude depending on the loading direction. This may have a large influence on the flow in (at least) some regions of the mantle [CHR 87], as was also shown for the flow of ice in ice sheets [MAN 97], but the topic has received little attention [BLA 07]. The key of this issue is to understand the link between single crys-tal rheology, microstructure (in particular LPO) and associated polycrystal behavior,e.g. as attempted for polar ices [CAS 08b].