Unfortunately not :-(

However, if you happen to be a travel agent, you can sign up for cxagents.com

Depending on your region, CX publishes their fare sheets in PDF format on this website so that travel agents in certain markets can follow the instructions to issue tickets for their clients (and even suggest the markup that the travel agent should add to the ticket). These are for fares that are not published in the GDS and may be lower/higher and have more restrictions (eg. no combinations with other fares, no interlining etc.) than the GDS fares. Certain online travel agencies also use these fare sheets to program into their system so that they can sell these fares to their customers online (I know that the UK, France versions of Expedia do, as well as the Italian AMEX agency mentioned earlier by the OP).

Note: AFAIK, the US and Canadian versions of the cxagents.com website don't have any faresheets as all the market fares ex-Canada / ex-US are distributed in the GDS (except fares targeted to and viewable only to travel agents who serve certain ethnic communities).

Here is a link showing an example of what a fare sheet looks like. (This is an old one I found by googling the Tour Code on my cousin's CX e-ticket)

http://file.fourseastravel.com/Qmult...13-28Feb14.pdf

If you walk into a travel agency in Asia, the travel agents are more likely to refer to a binder on their desk with a bunch of fare sheets (or look at PDF files such as the one above) as opposed to the GDS (unless they have them programmed in as "private fares") when you want to ask them for a fare quote.