While travelling between two Spacer planets, a ship has made a short stop at Earth. A dispute between two mathematicians, Alfred Ban Humboldt and Gennao Sabbat, has occurred, and R (obot) Daneel Olivaw has requested a chance to consult with Detective Elijah Baley on the matter.
Dr Humboldt has been thinking about a mathematics problem (one that Daneel admits he cannot understand/explain to Lije) and came to an epiphany for a specific technique in neural pathways. While aboard, he met Dr Sabbat and exchanged ideas, testing the concept against the well-respected peer's opinions to make sure they had not overlooked a flaw. Dr Sabbat was effusive with praise and, thus encouraged, Dr Humboldt prepared a paper on his idea. His account is backed up by his personal assistant, R (obot) Preston.
Dr Sabbat agrees with Dr Humboldt's account, except in the reversal of names, with their own personal assistant, R (obot) Idda, supporting their story. Neither mathematician will agree to an interview by an Earthman or consent to the Psychic Probe. Detective Baley is only able to interview the robots from an impromptu television contact. From these short interviews, he is expected to come to conclusive proof for which of the two are lying and attempting to steal the credit from the other.
"Mirror Image" has been republished several times: The Best Of Isaac Asimov (1973), Science Fiction Masters Of Today (1981), The Complete Robot (1982), Tin Stars (1986), The Asimov Chronicles: Fifty Years of Isaac Asimov (1989), Robot Visions Collection (1990), and The Complete Stories, Volume 2 (1992). It has also been published in Espace vital, the French translation of Earth is Room Enough.
"Mirror Image" contains examples of (beware! Unmarked spoilers below!):
- Androids and Detectives: Detective Baley and R (obot) Olivaw. In this story, the crime is Plagiarism; two mathematicians attempt to claim ownership over the same mathematical technique/proof. They are expected to find a solution from"Of these two men of great reputation, one is trying to destroy that of the other. By human values, I believe this may be regarded as worse than physical murder."
- Conviction by Counterfactual Clue: R (obot) Olivaw catches the In-Universe counterfactual when Detective Baley claims that the robot that suffered a Logic Bomb when asked to admit they had been lying is proof that they had been ordered to lie, instead of being the robot that had not been ordered to lie. Baley admits that, because he isn't a robopsychologist, either one is probably just as likely. His goal wasn't to accuse the owner of the robot who broke down, but to use whichever robot broke down to accuse the older mathematician of trying to commit Plagiarism.
- Casual Interstellar Travel: The Spacers are on a short (a few weeks) trip between Spacer solar systems, and stop by Earth because it isn't out of the way and Detective Baley might be able to resolve a problem the captain has been given.
- Framing the Guilty Party: When R. Preston shuts down, Baley takes that to mean the robot's owner is the guilty one, due to the strain causing a Logic Bomb. However, only a skilled roboticist would have known the guilty man's robot was under the bigger strain, and Baley is by no means one. He merely needeed some clear enough fact to put before the person he knew from the start was guilty.
- Game of Chicken: The plot is centered around two mathematicians aboard a starship accusing each other of stealing an important discovery. Should they make it to their destination (and therefore, an official investigation) with the matter unresolved, both will have their reputations ruined. Therefore, the innocent one is as likely to admit guilt as the guilty, and the guilty is likely to admit it in such a fashion that he will look innocent. Elijah actually calls it a game of intellectual chicken.
- Logic Bomb: Detective Baley manages to cause R (obot) Preston to shut down due to a conflict in the Three Laws during the interview. Because R (obot) Idda didn't break down during the same point, he takes this asymmetry of evidence as proof that R. Preston's owner was the plagiarist who stressed the Second Law, ordering their robot not to betray the truth.
- Long-Lived: Detective Baley reacts poorly to the statement that one of the mathematicians in this case is in his twenty-seventh decade, resenting their extended lifespans and advanced medicine since he is "not yet fifty" and that's the description of the young Spacer.
- The Namesake: The title refers to the way both mathematicians claim the same facts, but reverse ownership and actions, like a mirror reverses left and right.
- The Perry Mason Method: The perpetrator sweating happens off-screen by the ship's captain, but the opportunity is created by Detective Baley after interrogating the robots of both suspects, causing the guilty party's robot to undergo a Logic Bomb due to a Three Laws conflict. He's informed that the guilty party confessed quickly when the evidence of the broken robot is provided.
- Robot Names: The tradition established in The Caves of Steel continues: the robotic personal assistants are named R. Preston and R. Idda, with the "R" standing for "Robot".
- Three Laws-Compliant: The Three Laws are cited at the start of the work to ensure readers are familiar with the rules. In this story, one of two robots has been strongly ordered (Second Law) to keep something a secret due to how it would harm their master (First Law). Both robots give the exact same answers to questioning and Detectives Baley and Olivaw have to find some asymmetry in their otherwise "mirror" responses.
- Title Drop: The title is used to refer to the way both Dr Humboldt and Dr Sabbat tell the same story, but with the names of each involved reversed.
- Video Phone: Detective Baley is expected to resolve a conflict between two Spacer mathematicians by interviewing their personal assistants, R (obot) Preston and R (obot) Idda. Daneel has a micro-receiver and projector that he uses to turn a nearby wall into a video display, since their owners would never allow them to set foot on Earth.