We introduce a novel variant of seismic tomography that is based on colocated measurements of rotational and translational ground motions. Our aim is to assess whether rotations may be incorporated successfully into seismic inverse problems to produce better resolved and more realistic tomographic images. Our methodology is based on the definition of apparent S-wave speed as the ratio of rms velocity and rotation amplitudes. The principal advantages of this definition are that (1) no traveltimes measurements are needed and (2) the apparent S-wave speed is independent of source magnitude and source timing. We derive finite-frequency kernels for apparent S-wave speed by using a combination of the adjoint method and ray approximation. The properties of these kernels as a function of frequency bandwidth can be illustrated along with their usefulness for seismic tomography. In multifrequency synthetic inversions, we consider local crosshole tomography and regional-scale earthquake tomography. Our results indicate that S-wave speed variations can be retrieved accurately from colocated rotation and translation measurements, suggesting that our methodology is a promising extension of conventional seismic tomography. Further, apparent S-wave speed can be used to increase vertical resolution in teleseismic tomography for local structures.