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Compact polarimetry for a synthetic aperture radar (SAR) system is reviewed. Compact polarimetry (CP) is intended to provide useful polarimetric image classifications while avoiding the disadvantages of space-based quadrature-polarimetric (quad-pol) SARs. Two CP approaches are briefly described, π/4 and circular. A third form, hybrid compact polarimetry (HCP) has emerged as the preferred embodiment of compact polarimetry. HCP transmits circular polarization and receives on two orthogonal linear polarizations. When seen through its associated data processing and image classification algorithms, HPC's heritage dates back to the Stokes parameters (1852), which are summarized and explained in plain language. Hybrid dual-polarimetric imaging radars were in the payloads of two lunar-orbiting satellites, India's Earth-observing RISAT-1, and Japan's ALOS-2. In lunar or planetary orbit, a satellite equipped with an HCP imaging radar delivers the same class of polarimetric information as Earth-based radar astronomy. In stark contrast to quad-pol, compact polarimetry is compatible with wide swath modes of a SAR, including ScanSAR. All operational modes of the SARs aboard Canada's three-satellite Radarsat Constellation Mission (RCM) are hybrid dual-polarimetric. Image classification methodologies for HCP data are reviewed, two of which introduce errors for reasons explained. Their use is discouraged. An alternative and recommended group of methodologies yields reliable results, illustrated by polarimetrically classified images. A survey over numerous quantitative studies demonstrates HCP polarimetric classification effectiveness. The results verify that the performance accuracy of the HCP architecture is comparable to the accuracy delivered by a quadrature-polarized SAR. Four appendices are included covering related topics, including comments on inflight calibration of an HCP radar. [ABSTRACT FROM AUTHOR]