The concept of reflection can be extended to three-dimensional objects, including the inside parts, even if they are not transparent. The term then relates to structural as well as visual aspects. A three-dimensional object is reversed in the direction perpendicular to the mirror surface. In physics, mirror images are investigated in the subject called geometrical optics. More fundamentally in geometry and mathematics they form the principal objects of Coxeter group theory and reflection groups.
In chemistry, two versions (isomers) of a molecule, one a "mirror image" of the other, are called enantiomers if they are not "superposable" (the correct technical term, though the term "superimposable" is also used) on each other. That is an example of chirality. In general, an object and its mirror image are called enantiomorphs.
If a point of an object has coordinates (x, y, z) then the image of this point (as reflected by a mirror in the y, z plane) has coordinates (−x, y, z). Thus reflection is a reversal of the coordinate axis perpendicular (normal) to the mirrors surface. Although a plane mirror reverses an object only in the direction normal to the mirror surface, this turns the entire three-dimensional image seen in the mirror inside-out, so there is a perception' of a left-right reversal. Hence, the reversal is somewhat misleadingly called a "lateral inversion". The perception of a left-right reversal is geometrically explained by the fact that a three-dimensional object seen in a mirror is an inside-out version of the actual object, like a glove stripped off the left hand and turned into a right-hand glove, but there is still some confusion about the explanation amongst psychologists. The psychology of the perceived left-right reversal is discussed in "Much ado about mirrors" by Professor Michael Corballis (see "external links", below).
Reflection in a mirror does result in a change in chirality, more specifically from a right-handed to a left-handed coordinate system (or vice versa). If one looks in a mirror two axes (up-down and left-right) coincide with those in the mirror, but  the third axis (front-back) is reversed.
If a person stands side-on to a mirror, left and right hands will be reversed directly by the mirror, because the persons left-right axis is then normal to the mirror plane. However, it is important to understand that there are always' only two enantiomorphs, the object and its inside-out image. Therefore, no matter how the object is oriented towards the mirror, all the resulting images are fundamentally identical (as Corballis explains in his paper "Much ado about mirrors", mentioned above).
In the picture of the mountain reflected in the lake (photograph top right), the reversal normal to the reflecting surface is obvious. Notice that there is no obvious front-back or left-right of the mountain. In the example of the urn and mirror (photograph to right), the urn is fairly symmetrical front-back (and left-right). Thus, no obvious reversal of any sort can be seen in the mirror image of the urn.
A mirror image appears more obviously three-dimensional if the observer moves, or if the image is viewed using binocular vision. This is because the relative position of objects changes as the observer's perspective changes, or is differently viewed with each eye.
Looking through a mirror from different positions (but necessarily with the point of observation restricted to the halfspace on one side of the mirror) is like looking at the 3D mirror image of space; without further mirrors only the mirror image of the halfspace before the mirror is relevant; if there is another mirror, the mirror image of the other halfspace is too.