We use spherical projection and lifting operators in Euclidean d-space to describe a general framework for a variety of integral transforms arising in geometric tomography. These operators will be applied to support functions and surface area measures of convex bodies and to radial functions of star bodies. We then investigate averages of lifted projections and show that they correspond to self-adjoint intertwining operators. We obtain formulas for the eigenvalues of these operators and use them to ascertain circumstances under which tomographic measurements determine the original bodies. This approach via mean lifted projections leads us to some unexpected relationships between seemingly disparate geometric constructions.