A mirror actually reverses front and back. The image of your right hand remains directly in front of your right hand. But because you think of the image as a rotation of yourself, you're led to think that left and right are reversed.
Richard Feynman explains this with his usual wit and smiles here.
The "mirror puzzle" is commonly stated: Why does a plane mirror reverse left and right, but not top and bottom? This question refers to the left–right reversal in the shape of the mirror image of an object as compared with the original object, i.e., the reversal of the left–right asymmetry viewed from two different coordinate systems, each of which is intrinsic to the object or its mirror image. The left–right reversal in this sense always happens in mirroring the object for which left and right can be defined, irrespective of the relative configuration of the object to the mirror. The reason can be explained as follows:
Mirroring reverses the direction perpendicular to the mirror surface. Thus, the mirror image of an asymmetric object becomes its enantiomorph (an example of enantiomorphic pairs is a pair of left and right hands). An enantiomorph is, or can be considered to have been, obtained by reversal in any single direction of an object ("orientation reversing" mentioned in the third answer; also equivalent to "space inversion" or "parity operation" in physics, in which all the three directions are reversed).
However, we can define the left–right direction of an object (or the mirror image) only after defining the top–bottom and front–back directions from the external view* of the object (or the mirror image). Thus, the top and front of the enantiomorph is always regarded as the same sides, in the external view, of the top and front of the original object, so that the direction reversed in the enantiomorph has to be attributed to the direction defined last, i.e., the left–right direction.
The key point lies in the nature of the definition of left and right. Explanations by Richard Feynman, Martin Gardner and many others missed this point. The method of Feynman and Gardner to make a comparison between you and your mirror image is nothing but the precedent determination of the top–bottom and front–back directions of the mirror image.
More detailed explanations can be found in the following references:
- M. C. Corballis, "Much ado about mirrors." Psychonomic Bulletin & Review, Vol. 7, pp. 163-169 (2000).
- T. Tabata and S. Okuda, "Mirror reversal simply explained without recourse to psychological processes." ibid. pp. 170–173 (2000).
- H. Yoshimura and T. Tabata, "Relationship between frames of reference and mirror-image reversals." Perception Vol. 36, pp. 1049–1056 (2007).* The external view is mostly the shape. However, for a street car with a front–back symmetric shape, for example, motion defines the front–back direction.