Since the appearance of the Triple Helix model of innovations various extensions of the model to higher dimensions: Quadruple, Quintuple and N-tuple helices, had been proposed. Systemness of growing number of innovation system patterns can be expected to need classification. Also of no less importance is the way in which the new dimensions are added. This paper argues that while Triple Helix can be formed in a one unique way the higher extensions can be composed in a number of possible ways, so that the classification can be made according their topological or symmetry structure. The present paper alongside with presenting one more Triple Helix extensions for the fourth dimension, which preserves the Triple Helix model rotational symmetry and comprises nonlinear interaction in the joint working of the University, Industry and Government institutional spheres with complement of an additional actor - Media, is a step to classification of helix innovation system models on the symmetry base.