Two-dimensional optical implementations of permutation elements are cu
rrently of interest as components in switching network systems. These
elements can reorder and change the spacing of a set of input channels
at an output plane. Here an implementation of such an element that us
es holographic gratings is presented and theoretically analysed. Each
individual grating is recorded in a rotated position with respect to t
he others. In this way arbitrary 2D interconnections are achieved at t
he cost of growing design complexity. The complexity of the exposure p
rocedure results from the wish for high diffraction efficiency at reco
nstruction with light of mixed polarization. The inclusion of the Fres
nel reflection coefficients adds further to the difficulty of the anal
ysis. Experimental and theoretical results are compared and found to b
e broadly in agreement. This information is then used to implement two
practical components.