What twinning is
Two or more crystals of the same species grow in a specific symmetrical relationship, sharing lattice points across a plane or axis. The relationship is fixed, not random.
Random intergrowth is not twinning. The defining feature is that the orientation relationship is lawful.
Common forms
Contact twins meet along a plane, as in the familiar swallowtail gypsum. Penetration twins pass through one another, as in staurolite crosses and fluorite cubes.
Repeated or polysynthetic twinning produces the fine parallel striations visible on plagioclase feldspar cleavage surfaces.
Why it matters for identification
Twinning is often characteristic of a species and can settle an identification on its own. Staurolite crosses and carlsbad twins in orthoclase are effectively diagnostic.
It also explains apparent contradictions, such as a crystal that seems to show symmetry its system forbids.
Spotting it on a specimen
Look for re-entrant angles, where the outline of a crystal turns inward instead of continuing outward. A single untwinned crystal has a convex outline; a re-entrant notch is strong evidence of twinning.
On cleavage surfaces, look for fine parallel striations that shift the reflection as you tilt the specimen. That is polysynthetic twinning, and on a feldspar it separates plagioclase from orthoclase in a second.