In this paper, an automatic classification approach for polarimetric covariance structure is derived and assessed. It extends the framework of Pallotta et al. "Detecting Covariance Symmetries in Polarimetric SAR Images" to the heterogeneous environment, where the pixels of the polarimetric image share the same covariance structure but different power levels. The Principle of Invariance is exploited to replace the original data with a suitable statistic whose distribution is independent of the scale factors. Then, the classification problem is formulated in terms of a multiple hypotheses test and solved by means of model order selection rules. The behavior of the newly devised classifiers is first assessed over simulated data also in compari...