I do agree that this is very un-Java. However, how would method overloading allow for the simple given example of shuffling a sequence?

    seq.shuffle(); // copy to a new seq and shuffle it
    seq.shuffle$(); // shuffle the original seq

These two methods should have the exact same signature. So using two different names is necessary. I'm not particularly fond of introducing $ as a substitute for !. That convention is not existent in Java, and is not at all obvious to anyone who hasn't studied a functional language.

I would probably solve this by simply making shuffle always in place, and add a copy constructor and/or copy method:

    // Copy and shuffle.
    (new Seq(seq)).shuffle();
    seq.copy().shuffle();

    // Shuffle in-place.
    seq.shuffle();