Statistical approaches to texture analysis and synthesis have largely relied upon random models that characterize the 2-D process in terms of its first- and second-order statistics, and therefore cannot completely capture phase properties of random fields that are non-Gaussian and/or asymmetric. In this paper, higher than second-order statistics are used to derive and implement 2-D Gaussianity, linearity, and spatial reversibility tests that validate the respective modeling assumptions. The nonredundant region of the 2-D bispectrum is correctly defined and proven. A consistent parameter estimator for nonminimum phase, asymmetric noncausal, 2-D ARMA models is derived by minimizing a quadratic error polyspectrum matching criterion. Simulations on synthetic data are performed and the results of the bispectral analysis on real textures are reported.