Using Java, how can I compare every entry in HashMap to every other entry in the same HashMap without duplicating comparisons?

If you are not careful, the cost of eliminating duplicates could higher than the cost of redundant comparisons for the keys at least.

You can order the keys using System.identityHashCode(x)

for(Map.Entry<Key, Value> entry1: map.entrySet()) {
   Key key1 = entry1.getKey();
   int hash1 = System.identityHashCode(key1);
   Value value1 = entry1.getValue();
   for(Map.Entry<Key, Value> entry2: map.entrySet()) {
       Key key2 = entry2.getKey();
       if (key1 > System.identityHashCode(key2)) continue;

       Value value2 = entry1.getValue();
       // compare value1 and value2;
   }
}

How about this solution:

String[] values = map.values().toArray(new String[map.size()]);
for (int i = 0; i < values.length; i++) {
  for (int j = i+1; j<values.length; j++) {
    if (values[i].equals(values[j])) {
      // ...
    }
  }
}