κάτοπτρον (katoptron, "KAH-top-tron") is the mirror, and it is a rarer word than the everyday things it reflects: 52 occurrences across 17 works. Its distribution is almost entirely philosophical and introspective — Plotinus's Enneads (19), Aristotle's De Insomniis, "On Dreams" (5), Diogenes Laertius (5), Plato's Timaeus (4), and, tellingly, Plato's first Alcibiades (3). These are not works about grooming. They are works about seeing, and about seeing oneself.
Only one lexicon entry attaches to the lemma in the record — LSJ (Liddell–Scott–Jones, 1940), which gives simply "mirror" — and no etymology pointer was matched, so the word's deeper history is a gap here. But the form is transparent Greek: κατ- "down/back" with the root of ὄπτομαι, "to see" — the thing you look into to see back. The Greeks who used it were interested less in the glass than in the paradox it stages.
The cited passages hold that paradox exactly. In Plato's Alcibiades the mirror is the image for self-knowledge: as the eye sees itself by looking into the pupil of another eye, so the soul must look into another soul — the mirror teaches that you cannot see yourself directly. In Aristotle's On Dreams the mirror is evidence about perception and its distortions. And in Plotinus the mirror becomes metaphysics: the material world is a κάτοπτρον in which higher reality is only reflected, never present. Aeschylus, earliest of the cited authors, already uses it figuratively — the record places κάτοπτρον in the Agamemnon, where a mirror stands not for a face but for how one thing reveals another.
If the mirror is the one thing that shows you yourself only by not being you, is what it returns a self or an image?