I do wonder about your need to justify this question to me as something of a "real" question. This one ranks just behind the zero degrees Kelvin question that was asked by a woman and her boyfriend (who should have been busy having sex instead of thinking about physics) in terms of the amount of research that needed to be done. If I am simply to serve as a reference guide then I might as well just pack it in and hand all of my clients off to the Good Doctor. There must be better things for me to be doing with my time.

But I like you Keith. I like your inquisitive nature, and I like the fact that you actually put some real thought into this question. You even have a theory!

Unfortunately, that theory is wrong.

First of all, I think it would look really silly for the handkerchief to be hanging down from on top of the helmet, for that would destroy the entire helmet aesthetic. You would lose the brim, and what good would that be? Besides, you really should just tuck the handkerchief under the helmet itself, first laying it down on your head to serve the dual purpose of absorbing the sweat from your head and neck and also to serve to protect the neck (the pith helmet was most popularized before the days of SPF 30 sunscreen -- Dr. Livingstone did NOT march into the jungle with a case of sun block).

Secondly, my extensive research led me to the web site of the African Trading Company. With experience all the way back to 1997, I can think of nobody else who might serve as an authority on Pith Helmet protocol. They even have an entire section dedicated to the pith helmet, including a brief history AND the following photograph (which I have copied here without permission):

As you can see, our model is wearing her helmet with the band facing forward. I am taking their word for it, and I feel that you should too.

Reference:
African Trading Company's pith helmet page