Assigning a unique representation to equivalent circular queues
Enumerate all possible rotations of the queue.  Take the lexicographically first of them.  Use this as your representative.  If you want a short index into a hash table, take the hash of that.  Then any two equivalent queues will get the same representative / same index.

If the queue has $n$ items, implementing this naively takes $O(n^2)$ time.  However, it can be done more efficiently.  When the contents of the queue are random, sorting the rotations lexicographically, using a string comparison operator that only compares as much of a prefix as necessary to find the first mismatch, will have running time $O(n \log^2 n)$.  If you want to get more fancy, you can do it in $O(n)$ time, using a suffix tree or suffix array; though the constant factors and implementation complexity probably won't make this worthwhile unless your queues are extremely large.