Generally their entire display is a forbidden love between two brothers, also known as twincest or Hitachiincest.
They were originally only prone to spending time with each other due to people never being able to tell them apart (and their parents were always too busy to pay attention to them). However, when Haruhi Fujioka came around they soon had another person to add into their little circle of trust.
They often play the "Which one is Hikaru?" game in order to stump their fellow students (the only way to tell them apart, so they think, is which way they part their hair which they change daily). However, in the anime an easier way to tell them apart is their voices.
Their trust issues originated from a maid who had worked for their parents who betrayed them. They had liked her because she would play with them, but upon stealing money from their family they caught her in the middle of the act in which she said, "No one will ever be able to tell you two apart!"
Hikaru Hitachiin is more notorious for causing trouble than his younger brother Kaoru. He also has been known to have a small crush on Haruhi Fujioka (though he does not know it, while Kaoru does).