I was hired to choreograph and stage a show. Do I own the rights to the choreography? Can the producer recreate my choreography and copy it?

Yes, your choreography can be protected under copyright law and you are the author and own the copyright unless you sign a written agreement "selling" your rights to someone else. Choreography is defined under the US copyright law as a composition and arrangement of dance movements and patterns usually intended to be accompanied by music. There is one requirement, thought, that your work work  must be "fixed in a tangible medium of expression" from which the work can be performed. This can be a writing or a video tape of the work, the requirement means that it must be stored on some physical medium. You can prevent the producer from recreating it, so long as you register it. To enforce your copyright, you must register it with the U.S. Copyright Office. You can find information on registering your copyright at www.copyright.gov.