the center of inversion
Imagine a single special point inside an object such that, for every atom, if you draw a straight line from that atom through the point and continue an equal distance out the other side, you land on an identical atom. That special point is the centre of inversion. It is the symmetry of a perfect 'through-the-middle' turnaround: every feature has a twin directly opposite, the same distance away. The capital letter S has it; the letter A does not.
In coordinates it could not be simpler: put the centre at the origin, and inversion sends every point (x, y, z) to (-x, -y, -z). In the international notation it is written 1-bar (a bar over the 1, read 'one-bar'); in Schoenflies it is i. Like a mirror, inversion is improper, it flips handedness, and 1-bar is actually the smallest rotoinversion axis (a 360 degree 'rotation' followed by inversion is just inversion).
Why it matters enormously: a crystal that has a centre of inversion is called centrosymmetric; one that lacks it is non-centrosymmetric. That single yes-or-no decides whether a crystal can be piezoelectric (squeeze it and it makes a voltage): only non-centrosymmetric crystals can. Diffraction is also blind to the difference between a structure and its inverse (Friedel's law), so X-rays alone tend to make everything look centrosymmetric even when it is not.
In the CsCl structure a chlorine at (0,0,0) has a caesium at the body centre (1/2,1/2,1/2); the midpoint of the two chlorines around it is a centre of inversion, so the crystal is centrosymmetric.
An inversion centre sends (x,y,z) to (-x,-y,-z): every atom has a twin straight through the middle.
Presence of an inversion centre is the single most consequential yes/no in crystal symmetry: it gates piezoelectricity, pyroelectricity, and which Laue class you belong to.