This really made me think of Face/Off, where Cage and Travolta are in the place with mirrors after the bust on the apartment. And yours is a really good question... I don't think it makes tons of sense either, but I guess the movie makers explanation would be something along the lines of, 'After years of persuing each other, they want to finally understand each others motivation and thoughts before the bad guy is brought to justice.' or some crap like that.
