Intermezzo is narrated in a three-act structure that switches between Ivan, Peter, and Margaret's perspectives. In Part 1, Peter fixates on how horribly dressed his younger brother Ivan was at their father's funeral. This introduces their fraught relationship. Peter goes to his younger girlfriend Naomi's residence and informs her about his father's death. Peter supports Naomi financially and legally, promising to help decipher an eviction notice that she and her friends received. Later, he has dinner with his former partner Sylvia, who is still very present in his life. Peter spends his free time between the two women. Meanwhile, Ivan meets a woman named Margaret at a chess event. Though she is 14 years older than him, they quickly bond and begin to form a romantic relationship. Their age gap is a conflict for Margaret, but not for Ivan. Peter and Sylvia grow closer until they are forced to confront the fact that they cannot return to the past. After a tragic car accident, Sylvia experiences debilitating chronic pain that makes it impossible for her to have penetrative sex. For this reason, she feels unable to carry on a fully realized relationship with Peter.
When Peter finds out that Naomi has been evicted and arrested, he goes to her aid. She ends up staying with him at his apartment, where they enjoy sex and domestic life together, despite Peter's revulsion at their uncommon situation. While at dinner with Ivan, Peter cannot help but criticize Margaret for wanting to be with a younger man. This leads Ivan to profess his lifelong hatred of Peter. Ivan storms out of the restaurant and blocks Peter's number. Upset, Ivan calls Margaret and arranges to come see her. They swim in the ocean and go out to dinner. Margaret voices her doubts about their relationship, but they agree to keep seeing each other.
In Part 2, Sylvia expresses concern for Naomi following her eviction and arrest. Peter continues seeing both women, who fulfill different needs of his. The Koubek brothers' mother, Christine, calls Peter and asks about his new girlfriend. Peter realizes that he takes Ivan's side against Christine in their complex family dynamic. Unknown to Peter (since Ivan blocked his number), Ivan spends quality time with Margaret. They grow closer until Ivan tells Margaret he loves her. Margaret feels the same way, but she remains wary of their age difference. Despite Ivan's clear dislike of Peter, Margaret somewhat relates to the older brother for making mistakes. As the weeks pass, Ivan falls into a routine of working enough to meet his needs, studying and practicing chess during the week, and spending weekends with Margaret. He eventually picks up his dog, Alexei, from Christine's house and moves into his father's old residence in Kildare.
Peter takes care of Sylvia after an episode of intense pain. They have a sexual encounter that means very different things to each. Peter, believing he and Sylvia will get back together, proceeds to break up with Naomi and move her into the Kildare house. Naomi understands Peter's love for Sylvia, but she prompts Peter to admit that he is in love with her (Naomi), too. For Sylvia, the hookup was pleasant, but does mark the resumption of their relationship as it was before the accident. Complications also arise for Ivan and Margaret. Margaret runs into the captain of the town chess club, who insinuates that he (and thus the whole town) knows about her affair with Ivan. Though Ivan reassures her that this means nothing, Margaret feels caught between her love for Ivan and other people's judgment.
In Part 3, Ivan runs into Naomi at the house in Kildare. Ivan is outraged to learn that Peter was also involved in an age-gap relationship, and Naomi discovers that she apparently was never important enough to Peter that he would tell Ivan about her. Naomi's informed comments about their family get under Ivan's skin, and he retreats to his room. Meanwhile, Peter deals with fallout after a fight with Sylvia. He heads to Kildare to see Naomi, but instead encounters Ivan. Their heated argument escalates into a physical altercation. Peter seeks refuge at Christine's house. He has dinner and spends the night there. In the morning, Peter arrives back at his apartment in Dublin to find Sylvia and Naomi there together, extremely concerned about him. He faints with shock, and they care for him. Later, both Naomi and Sylvia privately express liking the other, and suggest that the three of them continue an arrangement.
Peter attends Ivan's chess tournament, which would grant him the second of three qualifications needed to earn the title of International Master. There, he runs into Margaret, and they are both humbled and charmed by each other. Ivan wins the tournament, and he and Peter reconcile. They begin planning a possible joint Christmas with Peter, Ivan, Sylvia, Naomi, and Margaret all in attendance.
