The underlying duality is Victorian social duality. The transformation of English society realized in the rise of a new type of middle class as a result of industrialization led to a resultant transformation that was realized by increased crime, violence and immorality of all kinds due to the absence of the governing constraints of small communities that had censured behavior and encouraged religious and moral deportment. The counterpressure to this disintegration of the moral side of English society was the Crown's emphasis on moral purity that so epitomizes Victorian life, an image that fails however to show the underlying social duality between decay and pursuit of purity. This is the social and cultural background to Stevenson's Dr Jekyll and Mr Hyde.
Within the text there are many dualities that Stevenson develops and explores. The primary one, of course, is the duality within Dr. Jekyll that manifests itself in the dramatic duality between Dr. Jekyll and Mr. Hyde (Who should have stayed hidden, eh?). This is a struggle between good and evil, virtuous impulses and base impulses, humanity and inhumanity. Dr. Jekyll, ostensibly an honorable public servant through the good of medicine, struggles with darker impulses that he wants separate from his nobler impulses: He actively wishes to create a duality of his inner psychology. Once this desire is manifest, the greatest duality occurs in the persons of the Dr.'s self, rendered unconscious, therefore belying the possibility of effective manifestation of dual tendencies, and Mr. Hyde's self, a self which eventually overpowers Dr. Jekyll's self, further belying the possibility of the manifestation of independent dual tendencies.
Dr. Jekyll's name represents another level of duality. "Jekyll" is an imperfect rhyme with "jackal," an individual who "performs dishonest or base deeds as the follower or accomplice of another" (Random House Dictionary on Dictionary.com). This is perhaps the saddest duality of all because even while Dr. Jekyll had un-experimented integrity of self and humanity, his name indicates that he was already--and seemingly always had been--the "handmaid," so to speak, of that which was base and dishonest, which is confirmed by his own reasons for undergoing the experiments. This raises a question about the nature of humanity: is there within each person a possibility for choosing between humanity and inhumanity or is it only present in some, and if some, then whom?
This leads to another level of duality that is embodied in Mr. Utterson himself and reinforced in his relationship with Mr. Enfield. Stevenson's description of Utterson sounds, upon an informed second reading, a lot like a description of Mr. Hyde (albeit at Hyde's best...) with the exception that Utterson had something eminently human, "a beacon," shinning from his eyes, which sets him far apart from Hyde, the embodiment of inhumanity. Utterson is Stevenson's contrast to Hyde. Similarly, the relationship between Utterson and Enfield, who is another one with Utterson's traits plus sociability and adventuresomeness, underscores through this doubling of dark characteristics that humanity may dwell within dark natures, dramatizing the duality of humanity versus inhumanity.
Other dualities are the thematic issues of the dual aspects of freedom: Hyde's freedom versus Utterson's and Enfield's freedom; and the duality of integrity versus transformation, which dovetails with the freedom theme.