Two-dimensional system dynamics depends on matrix pairs that represent the shift operators along coordinate axes. The structure of a matrix pair is analysed according to its characteristic polynomial and to the traces of suitable matrices in the algebra generated by the elements of the pair. Necessary and sufficient conditions for properties L and P are provided by resorting to Hankel matrix theory. Finite memory and separable systems, as well as two-dimensional systems whose characteristic polynomials exhibit one-dimensional structures, are finally characterized in terms of spectral properties and traces.