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The concept of using high-order intermodulation (IM) products beyond the third order for the analysis of passive intermodulation (PIM) sources is introduced. Current PIM characterization techniques focus on measurements of the third-order IM product. An advantage of observing higher order IM is demonstrated by showing that two simulated models, which predict an identical level of third-order IM, exhibit clear qualitative differences in the higher order IM products. A low residual PIM measurement system is able to measure up to the 29th-order IM product. A full discussion of uncertainties in the measurement technique used is given. The measured high-order IM products of a coaxial connector are used to fit and verify a model for its current-voltage characteristic. This fit model, a hyperbolic tangent function, accurately predicts the behavior of the higher orders as a function of input power and matches the third order over the 21-dB input power range with a root mean squared error of 3.5 dB. For another device, a microwave circulator, higher order IM products are used to deduce a model, which is then confirmed using a traditional measurement of third-order IM. High-order IM products are measured and analyzed in order to aid investigations of the physical processes causing PIM. Index Terms--Intermodulation (IM) distortion, nonlinear measurements, nonlinear systems, passive intermodulation (PIM).