a direct versus opposite isometry

Stand in front of a mirror and raise your right hand: the reflection raises its left. Now slide or turn a photo of yourself on a table: it never swaps your hands. That is the whole distinction. Some isometries keep the 'handedness' of a figure — its sense of clockwise versus counterclockwise — and some swap it. The first kind are called direct (or orientation-preserving), the second opposite (or orientation-reversing).

Concretely, label the vertices of a triangle A, B, C and note whether reading them goes counterclockwise or clockwise. A direct isometry leaves that reading unchanged; an opposite isometry reverses it. Among the four plane isometries, translations and rotations are direct — you can act them out by sliding and turning a cut-out without ever lifting it off the table. Reflections and glide reflections are opposite — they require flipping the cut-out over, swapping its front and back. The three-reflections theorem ties this to counting flips: an even number of reflections (zero or two) is direct, an odd number (one or three) is opposite. Composition follows a simple parity rule, just like multiplying signs: direct times direct is direct, direct times opposite is opposite, opposite times opposite is direct.

The distinction matters because a figure and its mirror image, though congruent, may not be the 'same' for practical purposes: a left shoe cannot be slid onto a right shoe within the plane, no matter how you translate and rotate it, precisely because that would require an opposite isometry. In chemistry the same idea names molecules that are mirror images but not superimposable (chirality); in everyday life it is why you cannot wear a left glove on your right hand. Direct isometries form a subgroup (composing two never escapes 'direct'), whereas the opposite ones do not, since two opposites compose to a direct.