Having a Multimap sorted on keys only in Java

Multimaps.index returns an ImmutableListMultimap, so you wouldn't be able to sort it after creating it. You could, however, first create a sorted copy of your Iterable<MyObject> and feed that to Multimap.index... ImmutableListMultimap keeps things in the same order it was given them.

public static ImmutableMultimap<Integer, MyObject> indexOnScore(Iterable<MyObject> i) {
  List<MyObject> sorted = Ordering.natural().onResultOf(myObjectToScore())
      .sortedCopy(i);
  return Multimaps.index(sorted, myObjectToScore());
}

Another option might be to create a TreeMultimap and use Ordering.arbitrary() as the Comparator for the values.

MultimapBuilder was introduced in Guava 16:

<K extends Comparable<? super K>, V> ListMultimap<K, V> multimap() {
    return MultimapBuilder.treeKeys().linkedListValues().build();
}

That keeps your keys sorted by their natural order (MultimapBuilder::treeKeys is also overloaded to accept a custom comparator), and the values associated with each key are maintained in a LinkedList (ArrayList and HashSet are among the other options).

Though the OP's specific situation seems to have been answered using immutable multimap building functions, I needed a mutable version of what he was asking for. In case it helps anyone, here's the generic method I ended up creating:

static <K, V> Multimap<K, V> newTreeArrayListMultimap(
    final int expectedValuesPerKey)
{
    return Multimaps.newMultimap(new TreeMap<K, Collection<V>>(),
        new Supplier<Collection<V>>()
        {
            @Override
            public Collection<V> get()
            {
                return new ArrayList<V>(expectedValuesPerKey);
            }
        });
}