Even with their limitless quarrels, their predestined tendency to always be at odds, they were inexplicably drawn to each other.
Some would call it narcissism, their habit of admiring each other (always reluctantly, always silently, their words inaudible, their glances unseen), but they knew better. As Katherine put it one night, when Elena expressed the worry after long hours of trading strange kisses and harsh comments, they weren't the same person. Of course, they looked exactly alike – perfect mirror images, standing side-by-side; but they weren't the same. Damon used to tell people (in an amused tone that could easily have been mistaken for contempt) that while the two were exact replicas of each other, physically identical down to the last strand of hair, their personalities were so obviously contrasting that anyone who knew them could always tell which one was Elena Gilbert and which one was Katherine Pierce.
In the old days, Elena would tentatively suggest that they proceed with caution, anxious that outsiders would make their nightly meetings impossible. But, as Katherine so shrewdly pointed out, everything about their existence was impossible, anyways, so why not add a little to the fun? They were the doppelgangers whose existence was an accident, at best; a supernatural fluke that had been born as a result of some witch's oversupply of cowardice and self-love. Maybe that's why Katherine hated everyone with such a passion, why Elena refused to run life-threatening risks; maybe a bit of their creator still lingered in them.
Katherine could grump and whine as much as she pleased, but Elena refused to see things in a weakened half-light. If they were impossible, if they couldn't, by any rule of the constant universe, even exist, then what was there to stop them? Their impossibility only gave way to potential circumstances that would normally be inconceivable. They were capable of anything, she pointed out, forcing Katherine to glance away from her sweat-plastered curls and into her eyes. The entire universe, every facet of every world, was open to their perusing, and even Katherine couldn't deny herself an adventure of that nature.
So in the end, Katherine realized, tracing patterns into the heated skin of her doppelganger, there was nobody to stop them. Screw conformity; their world had no limits. Maybe that's why they drew each other in; maybe, in a world that had no confines, they had to search for something solid, be it stable or unpredictable, to grasp onto.
Maybe it was why Katherine couldn't force herself away from Elena and why, when Katherine appeared in her window every night, informing her with glittering eyes that she was coming in come hell or high water, Elena only shrugged, and let her in.