This study investigates eigenvalue sensitivity to model parameters and veering in gyroscopic systems. The eigenvalue perturbation approach is formulated such that the results apply to discrete, continuous, and hybrid discrete-continuous gyroscopic systems. Third-order perturbation approximations for distinct eigenvalues are determined. Perturbations through second-order are derived for degenerate eigenvalues. The perturbation results are applied to a high-speed planetary gear model, where the results are shown to be accurate over a wide range of rotation speeds. The sensitivity of the eigenvalues to model parameters are written in terms of modal kinetic and potential energies. From the second-order perturbation approximation an eigenvalue veering parameter is defined and used to analyze veering in high-speed planetary gears, which is prominent in planetary gears that have disrupted cyclic symmetry.