The Radon cumulative distribution transform (R-CDT) exploits one-dimensional Wasserstein transport and the Radon transform to represent prominent features in images. It is closely related to the sliced Wasserstein distance and facilitates classification tasks, especially in the small data regime, like the recognition of watermarks in filigranology. Here, a typical issue is that the given data may be subject to affine transformations caused by the measuring process. The aim of this paper is to make the R-CDT and the related sliced Wasserstein distance invariant under affine transformations. For this, we propose a two-step normalization of the R-CDT and prove that our novel transform allows linear separation of affinely transformed image classes. The theoretical results are supported by numerical experiments showing a significant increase of the classification accuracy compared to the original R-CDT.