This article analyzes the global invariant properties of a class of exactly solvable area-preserving mixing transformations of the two dimensional torus. Starting from the closed-form solution of the expanding sub-bundle, a nonuniform stationary measures μw (intrinsically different from the ergodic one) is derived analytically, providing a concrete example for which the connections between geometrical and measure-theoretical approaches to chaotic dynamics can be worked out explicitly. It is shown that the measure μw describes the nonuniform space-filling properties of material lines under the recursive action of the transformation. The implications of the results for physically realizable mixing systems are also addressed.