One place where a left-handed version would definitely make sense is ligatures. It must be a bugger to tighten one up with the wrong hand.

I can't swap hands on a whistle at all (though I can switch between different fingering systems like clarinet, sax, flute, ocarina or recorder in a second). One of the most spectacular party tricks I've heard of a piper doing was John D. Burgess in a competition. He asked the judges after he'd won whether they'd noticed anything unusual about his performance. They hadn't watched carefully enough when he turned round away from them at one point. While playing a high A (no fingers) he'd switched hands on the chanter and carried on to the end that way.