We use a principal component analysis to characterize the finite-frequency sensitivity of seismic observables to anisotropy. A general anisotropic medium may be described in terms of 21 independent elastic parameters, each of which has an associated ‘primary’ sensitivity kernel. Our principal component analysis ranks linear combinations of the primary kernels to ascertain the dominant anisotropic parameters associated with a particular seismic observable. The principal parameters are those to which a given data set is the most sensitive. We demonstrate the efficiency of the method for a single arrival associated with a particular source–receiver combination, and apply it to a small synthetic Love-wave data set with a simple source–receiver geometry. For direct body wave arrivals, such as P, S and SKS, and direct Love and Rayleigh surface waves, our principal component analysis finds the same small combinations of dominant anisotropic parameters previously identified based upon asymptotic methods. The analysis further confirms the importance of mode coupling in finite-frequency surface wave sensitivity kernels. Our approach can be directly incorporated into a tomographic inversion to automatically select the general anisotropic parameters which are best constraint, for example, without prescribing the model to be transversely isotropic with a particular symmetry axis. The computational overhead associated with the calculation of the 21 primary kernels and the subsequent principal component analysis is minimal relative to an isotropic calculation.