Direct shear tests on rocks under compression-shear stress conditions have been widely conducted, whereas few have been performed under tension-shear stress conditions. However, rocks exhibit tension-shear failures in many scenarios, such as in the excavation disturbed zone in deep underground caverns and high slopes. A series of direct shear tests were performed with cuboid sandstone specimens under different normal tensile stresses (sigma(n) = - 3, - 2, and - 1 MPa) and compressive stresses (sigma(n) = 1, 3, and 5 MPa) at different shearing rates (v = 0.2, 1, 5, and 10 mm/min). The tension-shear tests were performed using an auxiliary device in combination with a compression-shear testing machine. The results showed that the fracture, shear stress-displacement curve, shear stiffness and shear strength were affected by both sigma(n) and v, and the differences in these mechanical behaviour between compression-shear and tension-shear cases were analysed in detail. The shear strength had a nonlinear relationship with both sigma(n) and v in the full region of tested normal stress (namely, the normal stress range from tension to low compression). The Hoek-Brown criterion (tau=A(sigma n-sigma t)B) considering the shearing rate effect (the relationship between parameter A (B) and shearing rate v is represented by a natural logarithm function) was proposed as the optimal shearing rate-dependent strength criterion for sandstone in the tested normal stress range.