Sherlock’s Mirror Image
As Sherlock, Irene, and Mycroft sit in the elegant dining room to negotiate, Sherlock sits quietly and patiently while Irene states her demands. Once Sherlock figures out what he wants to say and how to say it, he is quickly shot down and equally refuted by Irene. This battle goes on as each of them strike observational blows at each other until finally, Sherlock proves his point and silences Irene. These battles between Holmes and Adler occur frequently and fiercely throughout the novel and the film, with each character holding their own victories from time to time.
The concept of battling between Sherlock and Irene was relevant from the outset of their relationship. As Sherlock searches for something to wear for their first meeting, Irene picks out her “battle dress” to combat that. The back and forth action between each of the two individuals continue on throughout their interactions with each other, despite what was happening around them, perhaps suggesting that they may not be as individual as they seem.
Irene, in faking death, leads Sherlock to believe that he has won the battle since he now possesses the phone with all of the information on it. However, Irene outsmarts him by faking her own death and coming back to take back the phone. In another instance, Sherlock feels that he has outsmarted Irene yet again when he analyzes the email and figures out the plot to destroy the plane, however, he is simply playing into yet another trap set by the cunning Irene Adler.
While it may seem like the tension between Sherlock and Irene is far too strong for any connection to be made, the frequency and intensity of these back and forth battles of with and the relative evenness of the competitors show how each of them, in essence, are mirror images or equals to themselves.
I agree with your idea that the back and forth between the two reveals their true relationship, but you could go further and maybe explain why they “battle” so much. Is that Sherlock’s only way of showing affection?
Also, on a more inappropriate note, what else goes “back and forth?” Sex. Irene’s wit is the feature to which Sherlock is most attracted, so maybe their intellectual “back and forths” are supposed to take the place of actual, physical, sexual relations.