Non-imaging monostatic laser polarimetry has been used in a number of scenarios to probe characteristics of both surfaces and intervening media. While the measurement technology required for laser polarimetry has matured, sophisticated data-processing algorithms have been relatively slow to develop; hence laser-polarimeter data has been typically under-utilized. This paper presents systematic applications of components analysis to laser-polarimeter data that distinguish among electromagnetic-wave scattering characteristics of materials and enable the development of adaptive discrimination and monitoring algorithms that are invariant to selected variables in a scene. Both principal-components analysis (PCA) and non-linear components analysis are used to derive orientation- or pose-invariant channels from Mueller matrices measured over all probe angles. Invariant channels trained by using data due to isotropic scatterers are then used to conduct blind monitoring, i. e., predicting the presence of the target in a scene of arbitrary orientation, with the resulting cluster diagrams presented with photos of the illuminated scene components. Training of a monitor invariant over dual variables is demonstrated using data due to anisotropic scatterers.