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Rate-distortion analysis is applied to the problem of joint compression and classification. A Lagrangian distortion measure is used to consider both the Euclidean error in reconstructing the original data as well as the classification performance. The bound is calculated based on an alternating-minimization procedure, representing an extension of the Blahut-Arimoto algorithm. This rate-distortion framework is then applied to a joint compression and target-orientation estimation problem, based on a sequence of scattered waveforms measured at multiple target-sensor orientations. A hidden Markov model-Markov model (HMM-MM) is used as the statistical description for the source, here representative of multiaspect scattering data. Target-orientation estimation reduces to assessing the underlying HMM states from a sequence of observations. After deriving the rate-distortion function, we demonstrate that discrete HMM performance based on Lloyd encoding is far from this bound. Performance is improved via block coding, based on Bayes vector quantization. Results are presented for multiaspect acoustic scattering from an underwater elastic target, using measured and synthesized data.