The stable structure of a double-wall carbon nanotube (DWNT) is calculated for various chirality pairs, (n, m) - (n', m'), of inner and outer constituent layers. The stability of a double-wall nanotube is found not to depend on chirality, but rather on the diameter difference between inner and outer layers. However, the potential barrier for the relative displacement of the inner and outer nanotube layers is found to depend significantly on the chirality difference of the pair. Mechanical motions like a bolt-nut pair or discrete rotations can be expected for special pairs of chiralities in double-wall nanotubes, and these special motions will be important for nano-technology.