The coupling between the disk and spindle vibration modes of a rotating disk-spindle system is analyzed through the free vibrations of a rotating, flexible spindle with N attached flexible disks. The spindle is modeled as an extensible Kirchhoff-Love rod and the disks as Kirchhoff plates. Couplings between the longitudinal, torsional and flexural deformations of the spindle and the transverse and in-plane motions of the disk are studied analytically. A kinematically rich model captures couplings that have not been predicted previously. Discretization of these modes as a series of orthonormal functions allows for the construction of the characteristic matrix. The structure of this matrix is exploited to partition the eigenvalue problem into six natural classes and to provide simple, exact rules governing the coupling between the modes of the disk-spindle system. The longitudinal spindle vibration modes and the zero nodal diameter transverse disk modes are coupled inertially at all rotation speeds. The torsional spindle modes couple to the zero nodal diameter in-plane disk modes at all non-zero rotation speeds. This coupling is absent in a stationary disk-spindle system. For non-zero rotation speeds, the flexural modes of the spindle in the two orthogonal planes containing the undeformed spindle centerline and the one nodal diameter transverse and in-plane disk modes couple. The one nodal diameter transverse disk modes couple to the one nodal diameter in-plane disk modes through the flexural compliance of the spindle; this coupling cannot be observed through study of the disk alone.