It is shown that, although the shifts of Raman lines observed when random or uniaxially oriented polymers are uniaxially stressed depend on the polarization directions of the incident and scattered radiation, the shifts are not all independent. For a random sample, there is a very simple relationship between the four measurable shifts for small stresses, which is independent of the nature of the polymer or of the vibrational mode involved, so that only three of the shifts are independent. If the form of the Raman tenser is known, a further relationship exists between the shifts, so that only two of them are independent. For a uniaxially oriented polymer, the products of the shifts and corresponding intensities are related in such a way that, if the form of the Raman tenser is known, only three of the four measurable values are independent. The predictions are compared with experimental results on poly(ethylene terephthalate) and are confirmed within experimental uncertainty. For a random sample, it is possible to deduce the form of the function f(theta) that describes the average contribution to the shift produced by a chain making angle theta to the axis of stress. This function is found to have a form somewhere between that expected if the shift were proportional to the resolved stress parallel to the chain and that expected if the shift were proportional to the resolved strain.