The enumeration of Hamiltonian cycles on 2n*2n grids of nodes is alongstanding problem in combinatorics. Previous work has concentrated oncounting all cycles. The current work enumerates nonisomorphic cycles -- thatis, the number of isomorphism classes (up to all symmetry operations of thesquare). It is shown that the matrix method used previously can be modified tocount cycles with all combinations of reflective and 180-degree rotationalsymmetry. Cycles with 90-degree rotational symmetry were counted by a directsearch, using a modification of Knuth's Dancing Links algorithm. From thesecounts, the numbers of nonisomorphic cycles were calculated for n<=10.
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