Euler angles (α,β,γ) are cumbersome from a computational point of view, and their link to experimental parameters is oblique. The angle–axis {Φ,n^} parametrization, especially in the form of quaternions (or Euler–Rodrigues parameters), has served as the most promising alternative, and they have enjoyed considerable success in rf pulse design and optimization. We focus on the benefits of angle–axis parameters by considering a multipole operator expansion of the rotation operator D^(Φ,n^), and a Clebsch–Gordan expansion of the rotation matrices DMM′J(Φ,n^). Each of the coefficients in the Clebsch–Gordan expansion is proportional to the product of a spherical harmonic of the vector n^ specifying the axis of rotation, Yλμ(n^), with a fixed function of the rotation angle Φ, a Gegenbauer polynomial C2J-λλ+1(cosΦ2). Several application examples demonstrate that this Clebsch–Gordan expansion gives easy and direct access to many of the parameters of experimental interest, including coherence order changes (isolated in the Clebsch–Gordan coefficients), and rotation angle (isolated in the Gegenbauer polynomials).