"This is a story about truth, beauty, freedom: but above all things, this is a story about love." --Christian, writing on his typewriter.
This story has its Orphean elements to be sure, but there is more here than Greek mythology. For starters, Orpheus' love is ultimately selfish, Christian’s and Satine's is not. Satine is changed by love and ultimately is willing to sacrifice herself to save Christian's life. Also Satine dies not due to some fateful wish of the gods, but due to a life in the environment of the Moulin Rouge that ultimately kills her.
The love in Moulin Rouge begins as fleshly love: flashy and enticing, but costly and fake. Then Christian and Satine discover selfish romantic love. This drives them further to a love that lasts, that is unselfish, even sacrificial. But they are both caught by their pasts. Christian fails to see beyond the surface of things and fails to see Satine’s true love for him. Satine in turn is forced by her past sins to reject him in an effort to save him. In the end, however, Love wins when Satine reveals her love for Christian, dooming the Moulin Rouge and endangering herself. When she does finally die, lying on the theater floor, she commissions Christian to "tell their story" prompting the whole telling of the movie. The lesson learned?
"The greatest thing you'll ever learn is just to love and be loved in return." --Christian, again on the typewriter.
So the themes in Moulin Rouge are sin, sacrifice, regeneration, truth, beauty, goodness, and love. The story, like Orpheus, sets out into Hades to save Love, but unlike in the Greek tragedy, Love wins in the end.