Valuations on Lipschitz functions

We consider the space Lip(S^{n-1}) of Lipschitz functions defined on the unit sphere. We prove a characterization result for valuations on Lip(S^{n-1}) which are continuous and invariant under rotations and under the addition of linear functions. We also study valuations on Lip(S^1) which are just continuous and rotation invariant and we prove an integral representation for them. As a corollary, we obtain a characterization result for all uniformly continuous and rotation invariant valuations on Lip(S^1).