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This article presents and discusses in detail the results of extensive exactcalculations of the most basic ingredients of spin networks, the Racahcoefficients (or Wigner 6j symbols), exhibiting their salient features whenconsidered as a function of two variables - a natural choice due to theirorigin as elements of a square orthogonal matrix - and illustrated by use of aprojection on a square "screen" introduced recently. On these screens, shownare images which provide a systematic classification of features previouslyintroduced to represent the caustic and ridge curves (which delimit theboundaries between oscillatory and evanescent behaviour according to theasymptotic analysis of semiclassical approaches). Particular relevance is givento the surprising role of the intriguing symmetries discovered long ago byRegge and recently revisited; from their use, together with other newlydiscovered properties and in conjunction with the traditional combinatorialones, a picture emerges of the amplitudes and phases of these discretewavefunctions, of interest in wide areas as building blocks of basic andapplied quantum mechanics.

The Screen representation of spin networks. Images of 6j symbols and semiclassical features