Are Directors supposed to consult the Producer about their casting decisions before finalizing them?  Or do they just let the Producer know the choices when finished? 

Well the key phrase in your casting issue is that the part was "either/or" - which I assume you to mean that the show ALLOWS for using a male or female with the author's blessing and not violating the author's casting choices.  SO that being the case - if it is - then I would NEVER allow anyone to DICTATE who I cast, why or anything else UNLESS it is pre-defined as something the producing company, the producer or any other entity has allowed for and made known to me as director BEFORE they hire me and before casting begins.  If they let me know ahead of time that this is how they do things, and I accept (first of all I NEVER WOULD) then I have little or nothing to say about it.  BUT if they do NOT define this parameter, I make MY casting choices based upon what I want, then they try to come back and define how I should cast or make any kind of comments based upon what I've done and how they'd like it some other way - too bad!!!  I ALWAYS have and always WILL make my own casting choices without input from anyone (well except the author or copyright company).  I will not allow someone to dictate to ME what they hire ME to have the experience, expertise and know-how to do for the good of the production.

In the Pros, I still believe its the Producer who has the say... That being said..

The Producer ultimately represents the CT's board, along with its constitution, bylaws, policies, and procedures. 

That means that the Producer can and may overrule the Director on any artistic decision, including casting choice, IF the Director's artistic decision goes against the CT's procedures, policies, bylaws, or constitution. 

Remember that the Producer has the CT's long-term interests (into the past and into the future) at heart, whereas the Director probably only has this particular production uppermost in his mind.

If the CT would prefer that roles are filled by actors of the appropriate stated gender, then it is up to the Producer to communicate and enforce this policy with the Director, and in a timely fashion (i.e., BEFORE any actor is offerred the role), and would veto the Director's casting choice.

If the CT doesn't care which gender of actor plays any particular role, or if the author and/or publisher openly states that the role can be played by any gender of actor, then the Producer would probably not veto the Director's casting choice.

I suspect that in most CT's the Director has been given relatively free reign in casting, and the CT has gotten used to NOT interfering in casting decisions.  This in no way frees the Producer from the important responsibility to ensure that every Director's casting decisions comply with the CT's policies, procedures, bylaws, and constitution.

We're having a meeting today to discuss our misunderstanding of the creative control a Producer has.  We clearly have two different interpretations of the script, which we did not realize before they hired me.  I gave her the list of responsibilities I am used to working under and they did not change any of them.