One of the things Christopher Nolan has done amazingly well in his Batman Trilogy is to push along Batman's struggle with his identity and mission. This is, after all, the heart of any work of literature, the protagonist's inner struggle to come to terms with himself. As an Epic hero, however, this is not just Batman's struggle, but Gotham's too.
One of the chief ways in which Nolan has accomplished the task of pushing Batman to question himself is to confront him with characters which are mirror images of himself. Let's examine some of them:
Ra's al Ghul - Made Batman who he is. Nevertheless, as Batman himself says, there is something that makes Ra's al Ghul and his League of Shadows vigilantes while Batman remains a legitimate source of order and justice. That is the fact that Ra's al Ghul seeks to impose his own will on the world, his own idea of what's good for it, rather than freeing it up to do the right thing itself, first from the Mafia which had the city by its throat and then fromt he Joker's chaos. As Batman says, he hopes for the day when Gotham will no longer need him. As the opening sequence of The Dark Knight proves, his presence is preventative as well as punitive.
The Scarecrow - Uses fear to conquer his opponents. How is Batman different? The Scarecrow is a weak, calculating wimp who uses a gas to induce fear chemically. Batman is big and strong, and strikes fear into the heart of criminals by his very existence. Remember that in art, exterior attributes are never arbitrary, but illustrative of the interior realities of the characters. What this means here, I think, is that Scarecrow is a snake, a fraud. He uses a fake fear to get his way. Batman on the other hand, has more in common with the fear we should have for God. It is his very being which should be respected and looked upon in awe.
Commissioner Gordon - In one of the comic books that inspired Christopher Nolan, Frank Miller's Batman: Year One, you can almost split the pages up evenly between Batman and Gordon as you alternate perspectives on their parallel roles in the main plot and character development. Nolan has done the same thing in his films, drawing the comparison as often as possible. How they're alike? They are both keepers of the peace and ministers of order. How do they differ? Gordon operates inside the written law. Batman serves a higher law because he believes the lower law of men has failed.
Thomas Wayne - Like his father before him, Bruce Wayne seeks to work toward the good of his city and people. Unlike his father, Bruce got to witness first hand his father's lack of success. In Batman Begins, the Mob has made Gotham so afraid to go out that they won't use the train Bruce's father built for them, having left it to rot and decay from misuse and maltreatment. I think it's significant that the solution to impending disaster is the destruction of the train. Batman has established a new order, one in which such half-measures have been done away with.
Two-Face - Like Two-Face, Batman must play host to two personalities, one forever challenging the other for supremacy. Unlike Two-Face, Batman's decisions about who he wants to be (not to mention his decisions in general) are purely his own, settled on after thought and deliberation rather than the flip of a coin.
The Joker - "You're just a freak, like me." "You complete me." Notice how in the last Joker scene in The Dark Knight, Batman has him strung upside down. They are set in opposition to one another during their conversation. Yet, as it progresses, Nolan rotates the camera on the Joker so that he is upside-down from Batman's perspective, but cutting back and forth between the two of them, they appear to us as if they're both upright. It's a masterful visualization of how the two characters are a mirror for one another and also polar opposites.
The Catwoman - Of course we can't be exactly sure what Nolan's going to do with Catwoman, but we can at least say that she will mimic Batman's wits and cunning. The struggle with Catwoman is always her skewed intentions, viz., her own good rather than that of her city. She also threatens to seduce Batman into involvement in romantic affairs, which he must be bigger than.