Here is a way to generate the 80 permutations using PROC CLP in SAS/OR:

proc clp out=out allsolns;
var (X1-X6) = [1,6];
alldiff();
lincon
X1 ne 1, X1 ne 2,
X2 ne 1, X2 ne 2,
X3 ne 3, X3 ne 4,
X4 ne 3, X4 ne 4,
X5 ne 5, X5 ne 6,
X6 ne 5, X6 ne 6;
run;