This paper is devoted to modeling the asymmetry and anisotropy in yielding of pressure insensitive metals. First, a new isotropic yield criterion with threefold rotational symmetry about the origin is proposed. This criterion is an odd function in the principal values of the stress deviator. It involves only one parameter that is expressible in terms of the yield strengths in compression and tension. Comparisons with the results of polycrystalline simulations performed by Hosford and Allen [Met. Trans. 4 (1973) 1424–1425] show that the proposed macroscopic criterion describes very well the asymmetry in yielding due to activation of twinning. Further, the extension of this isotropic criterion to orthotropy is developed. The proposed anisotropic criterion is applied to the description of the plastic response of hexagonal close packed (hcp) materials. It is shown that the proposed anisotropic criterion reproduces with great accuracy both the higher strength in tension than in compression and the higher tensile strength in the transverse direction than in the rolling direction observed in textured magnesium and magnesium alloys sheets (data after Kelly and Hosford [Trans. TMS-AIME 242 (1968) 654–661]).