Why not allow an external interface to provide hashCode/equals for a HashMap?

With a TreeMap it's trivial to provide a custom Comparator, thus overriding the semantics provided by Comparable objects added to the map. HashMaps however cannot be controlled in this manner; the functions providing hash values and equality checks cannot be 'side-loaded'.

I suspect it would be both easy and useful to design an interface and to retrofit this into HashMap (or a new class)? Something like this, except with better names:

  interface Hasharator<T> {
    int alternativeHashCode(T t);
    boolean alternativeEquals(T t1, T t2);
  }

  class HasharatorMap<K, V> {
    HasharatorMap(Hasharator<? super K> hasharator) { ... }
  }

  class HasharatorSet<T> {
    HasharatorSet(Hasharator<? super T> hasharator) { ... }
  }

The case insensitive Map problem gets a trivial solution:

 new HasharatorMap(String.CASE_INSENSITIVE_EQUALITY);

Would this be doable, or can you see any fundamental problems with this approach?

Is the approach used in any existing (non-JRE) libs? (Tried google, no luck.)

EDIT: Nice workaround presented by hazzen, but I'm afraid this is the workaround I'm trying to avoid... ;)

EDIT: Changed title to no longer mention "Comparator"; I suspect this was a bit confusing.

EDIT: Accepted answer with relation to performance; would love a more specific answer!

EDIT: There is an implementation; see the accepted answer below.

EDIT: Rephrased the first sentence to indicate more clearly that it's the side-loading I'm after (and not ordering; ordering does not belong in HashMap).