Texture analysis based on textons is extended by introducing a method for computing textons of arbitrary order. First-, second- and third-order textons are applied to classify screening mammograms as to indicate a low or high risk of breast cancer. First-order textons are found to provide better estimates of breast cancer risk than other orders on their own but the combination of first- and second-order textons outperforms first-order textons alone and other combinations of two orders. Combining all three orders of textons does not improve classification. This example indicates that including higher-order textons has the potential to improve classification performance.