Differential cross section (DCS) measurements are reported for scattering of a He atomic beam by crossed beams of Ar, C2H2, CO2, and OCS. Relative to the HeAr diffractive structure, the HeC2H2 DCS is moderately damped at small scattering angles and severely damped at large scattering angles; the HeCO2 and HeOCS DCS’s are severely damped for all scattering angles. This damping directly reflects increasing anisotropy of the scattering partner from Ar (none) to C2H2 (moderate) to CO2 and OCS (strong). Even though the present data do not resolve elastic from inelastic contributions, the HeCO2 results are consistent with partially resolved inelastic DCS measurements [U. Buck, H. Meyer, M. Tolle, and R. Schinke, Chem. Phys. 104, 345 (1986)] and therefore complement them. Consequently the data are used to obtain empirical anisotropic intermolecular potentials for HeC2H2 and HeOCS within the infinite-order-sudden approximation. These reproduce the total (unresolved elastic +inelastic) DCS measurements very well, even though the angular asymmetry of OCS is ignored in the analysis.