Does Java have a HashMap with reverse lookup?

There is no such class in the Java API. The Apache Commons class you want is going to be one of the implementations of BidiMap.

As a mathematician, I would call this kind of structure a bijection.

In addition to Apache Commons, Guava also has a BiMap.

Here is a simple class I used to get this done (I did not want to have yet another third party dependency). It does not offer all features available in Maps but it is a good start.

    public class BidirectionalMap<KeyType, ValueType>{
        private Map<KeyType, ValueType> keyToValueMap = new ConcurrentHashMap<KeyType, ValueType>();
        private Map<ValueType, KeyType> valueToKeyMap = new ConcurrentHashMap<ValueType, KeyType>();

        synchronized public void put(KeyType key, ValueType value){
            keyToValueMap.put(key, value);
            valueToKeyMap.put(value, key);
        }

        synchronized public ValueType removeByKey(KeyType key){
            ValueType removedValue = keyToValueMap.remove(key);
            valueToKeyMap.remove(removedValue);
            return removedValue;
        }

        synchronized public KeyType removeByValue(ValueType value){
            KeyType removedKey = valueToKeyMap.remove(value);
            keyToValueMap.remove(removedKey);
            return removedKey;
        }

        public boolean containsKey(KeyType key){
            return keyToValueMap.containsKey(key);
        }

        public boolean containsValue(ValueType value){
            return keyToValueMap.containsValue(value);
        }

        public KeyType getKey(ValueType value){
            return valueToKeyMap.get(value);
        }

        public ValueType get(KeyType key){
            return keyToValueMap.get(key);
        }
    }

If no collisions occur, you can always add both directions to the same HashMap :-)