How does Wilde present the relationship between Cecily and Gwendolen in The Importance of Being Earnest?
The play intentionally makes double characters make mirrors of each other. For example, Jack and Algy are reflections of each other's thoughts and behaviors. Even Jack and Algy's butlers are reflective of each other.
Similarly Gwen and Cecily, though are from different social ranking, are still exact replicas of each others' frames of mind:
First, Wilde foreshadowed the relationship on Act I, when Jack told Algernon that the girls will be pleased to meet and that the relationship of Cecily and Gwen will grow until they call each other "sisters", to which Algernon replied: "That's after they call each other many other things"- which is what occured.
Second, Wilde puts similar dialogues on each female character: Both obsessed with the name Earnest, both idealizing love and marriage, and both stopping at nothing to get what they want. Also, both are not presented as sharp and brilliantly as Algy and Jack, both lack lustre as characters, and -as stated before- are reflections of each other.
Third: As both Gwen and Cecily had just become entangled to an "Earnest" (Jack and Algernon respectively) they thought to be each other's rivals. This is when Wilde dictates a witty showdown between the two, which discloses that, different social status or not, the two women are exact behavioral clones of each other.
Gwendolyn used rank, status,and sarcastic witticism to bring down Cecily, and Cecily used the same amount of witticism, sarcasm, and come backs to beat down Gwendolyn.
None of the women out did the other, for they are a representation of the Victorian female desperation and idealization of marriage that Wilde criticized so much.
In the end, they did call each other "sisters- after they would have called each other many other things", continued their insane obsession over the name Earnest, and ended up fulfilling their wishes by becoming engaged to these two men they hardly knew well but by name.