Are there any major commodity markets with a public order book? Most major exchanges seem to have secret order books.

For example, on the NYSE only the specialists know the order books.

On the NASDAQ only the market maker can see the order book.

On Globex, the full order book (Level 6) is kept secret.

Are there any major exchanges with a public order book?

Example of a public order book:

https://www.bitstamp.net/market/order_book/
Major trading exchanges have order books that anyone can see, if they're willing to pay for the service. So in that sense, they are available to the public. For example, anyone can subscribe via a broker to the London Stock Exchange Level 2 data, which shows them the live order books for listed instruments.

Very few have "secret" order books. I think you mean dark pools; they do exist too, and wikipedia has a list.