Throughout the novel doubles are used by Dickens to draw contrasting parallels. We can see this with the focus on the two cities, and the opening sentence clearly establishes the importance of doubles to the narrative.
As you have correctly identified, Dickens sets off various characters against each other, such as Carton and Darnay, but also Lucie and Madame Defarge, to highlight certain themes within the novel. It is clear that they are both complete opposites: Lucie is loving, kind, gentle and nurturing, as evidenced through the care of her father, just as Madame Defarge is consumed with revenge and with a desire for vengeance.
However, this pairing of opposites is used by Dickens to link in to the various themes of the novel. For example, it is Lucie's dedication, commitment and love of her father that makes his healing and renewal possible (linking in to the theme of resurrection), just as Madame Defarge's bloodlust and singleminded focus on achieving revenge against the aristocracy thus generates an endless cycle of violence and oppression.