Tensor target spin asymmetries T2M (M=0,1,2) in the reaction γd → π−pp are studied for photon energies from π-threshold up to 1.5 GeV with inclusion of rescattering effects. It is shown that the influence of rescattering effects on the tensor target spin asymmetries is sizable in the energy region near π-threshold. At higher energies, much smaller influence of rescattering effects is seen. The sensitivity of the obtained results to the elementary pion photoproduction operator is investigated and a considerable dependence is found, in particular at forward pion angles. In addition, a comparison with results of other theoretical models is also given. The extracted spin asymmetries are compared with available experimental data and a qualitative agreement is obtained. The predictions presented here may be useful to interpret the recent measurements from the VEPP-3 electron storage ring.