Katara found comfort in routine. She liked the clarity of order, and the simplicity of neatness. Her wild hair was meticulously braided twice a day, her clothes were never spotted, and she always began preparing a meal by arranging her cooking equipment like a surgeon's tools.
Zuko found peace in discipline. As a prince, his education was extensive, and the many lessons required strict scheduling. He meditated twice a day, his attire was never wrinkled, and he always began a meal by arranging the place setting into the closest possible imitation of a court dining table.
This is not to say that either of them was inflexible; both had learned to improvise from their experience on the road. How either of them remained tidy in such circumstances was a testament to their remarkable dedication to good hygiene.
Katara watched the prince during meals. She unconsciously imitated his posture and mannerisms. Zuko carried himself with elegance, even in the most informal situations. He made her reconsider the definition of posture.
Sokka thought good manners were, "girly" but Zuko ate with strong, masculine grace. When the young men sat next to each other, she was ashamed of what her brother's habits said about her tribe. She had come to think of Sokka's sloppiness as a character flaw and thought herself superior by contrast. That changed when Zuko joined the group. For the first time in her life, Katara felt like a peasant.
Zuko was very good at not being caught staring. He watched Katara when she walked, when she trained and even when she slept. He unconsciously imitated her fluid movements and unguarded expressions. She made him reconsider the meaning of good manners.
Zuko had known and admired (or feared) many strong women, but none had been possessed of such natural and untrained courtesy. He found her frank nature completely disarming. Katara was even honest when angry. He could not stop watching her; she was a creature of remarkable grace. Until he spent time with her, the prince had believed that his culture and class set him far above others. Now, for the first time in his life, Zuko felt uncivilized.
They were both humbled. They were both maturing. They were both falling deeply in love.