While I have already done a section on Catherine, this has nothing to do with what category their relationship falls under. As he tended to be towards everyone he met in the television series, Trowa was distant and to the point with everyone at the circus. He viewed the place as a convenience and kept things professional. Thus, when he becomes Catherine Bloom's assistant he remains distant despite her friendliness.
This behavior makes me think of how things likely were with the Mercenary Corp. he was raised by. On their down time the men were likely very chummy with him. When things became difficult to downright disastrous, however, they all certainly became very different men. As No-name, he likely witnessed plenty of betrayal where suddenly two men who seemed like brothers cared only for the motto, "it's every man for himself."
He obviously knew of this darkness by the time he was around ten when he met Midii Une. It could be possible that No-name had no reason to cry over his lost unit because he doubted any of them would have sincerely cried for him. It was a mercenary unit. When one man falls, he will eventually be replaced with another. He was easily replaceable.
Trowa's view of the world was likely "dog-eat-dog". People were quick to turn on each other in unfavorable situations. Catherine was a civilian at that. She would quickly flee and save herself. Thus, like an instilled mechanism, he ignored all her efforts to make him feel welcomed and get to know him. He dismissed even her efforts to apologize to him after his first performance with her. His mindset was in the only set it had ever been taught: do your duty, do not fail, you are responsible for yourself.
Thus we come to the event that many viewers apparently love and enjoy recreating in fan-fictions and fan-comics for any time Trowa does something disagreeable in Catherine's eyes: Trowa's self-destruction performance. I will not say the method was overkill when considering how Trowa’s mind likely works. A killer punch to the cheek would be necessary to stun him long enough to let Catherine air out her grievances and get through to him. I do, however, believe that punching a person would be wrong and disastrous in a real scenario dealing with an actual person dead-set on committing suicide.
Anyway, her point was driven through Trowa's head figuratively and literally. He hadn't even died, yet a person he considered as an acquaintance at best shed bitter tears over his attempt to leave his role as a Gundam Pilot without a cause via self-destruction. He likely remained stunned by this information for a while and took all night to process it. Nevertheless, actions speak louder than words and all of Catherine's rang strong and true.
Catherine Bloom's existence established to Trowa that his life really did matter to others. She did not see him as just a co-worker or another performer. He was not just another useful means to earn money from others. He was not easily replaceable. Even though he had no known family, his life would still matter to them. Everyone at the circus, especially Catherine, truly cared for him as an individual. He wasn't on the battlefield there. He was where he could belong by just being himself.