In the story, Kikuji was invited to the tea ceremony by Kurimoto Chikako, his father's (Mr. Mitani's) one-time mistress. Kurimoto invited Kikuji because she wanted him to meet a young lady who was taking ceremonial tea lessons from her. In reality, Kurimoto had arranged the miai (or prospective bride-viewing) to spite Mrs. Ota, who was also another of Mr. Mitani's mistresses.
Kurimoto's jealousy was predicated on the fact that Mr. Mitani had left her for Mrs. Ota when he was alive. Accordingly, Mr. Mitani reportedly kept Mrs. Ota as his mistress until the day he died, while Kurimoto was relegated to the background as a sexless but "convenient fixture." To act insult to injury (where Kurimoto was concerned), Mrs. Ota stayed youthful-looking and beautiful even as she, Kurimoto, struggled to embrace her ambiguous feminine identity.
When she made her invitation to Kikuji, Kurimoto understood that Mrs. Ota and her daughter would be in attendance. Nevertheless, Kurimoto reveled in her ability to upstage Mrs. Ota; she wanted to show her nemesis that she still had some influence over Kikuji's life. As if to indicate her spiteful intentions, Kurimoto used a specific tea bowl to serve Miss Inamura. Miss Inamura was the girl Kikuji was supposed to meet; she wore a thousand-crane scarf to the tea ceremony. The bowl itself had been initially owned by Mr. Ota. Upon Mr. Ota's death, the bowl was inherited by Mrs. Ota; she later gave the tea bowl to old Mr. Mitani (her lover). Mr. Mitani then gifted the bowl to Kurimoto Chikako. Essentially, Kurimoto's use of the tea bowl was a tactless move on her part.
So, the real reason Kikuji was invited to the tea ceremony had very little to do with courtesy. Kurimoto ostensibly invited Kikuji so that he could meet with Miss Inamura. However, Kurimoto's real intentions were to make Mrs. Ota jealous of the supposed influence she still has in Kikuji's life.