It is proved that if f is an operator Lipschitz function defined on ℝⁿ, then the function f∘φ/ǁφ′ǁ is also operator Lipschitz for every Möbius transformation φ with f(φ(∞)) = 0. Here ‖φ′‖ denotes the operator norm of the Jacobian matrix φ′ Similar statements for operator Lipschitz functions defined on closed subsets of ℝⁿare also obtained. Bibliography: 10 titles.