  The four-bar linkage is a basic arrangement of mechanical engineering and
represents the simplest movable system formed by a closed sequence of
bar-shaped bodies. Although the mechanism can have in general a spatial
arrangement, we focus here on the prototypical planar case, starting however
from a spatial viewpoint. The classification of the mechanism relies on the
angular range spanned by the rotational motion of the bars allowed by the
ratios among their lengths and is established by conditions for the existence
of either one or more bars allowed to move as cranks, namely to be permitted to
rotate the full 360 degrees range (Grashof cases), or as rockers with limited
angular ranges (non-Grashof cases). In this paper, we provide a view on the
connections between the "classic" four-bar problem and the theory of 6j symbols
of quantum mechanical angular momentum theory, occurring in a variety of
contexts in pure and applied quantum mechanics. The general case and a series
of symmetric configurations are illustrated, by representing the range of
existence of the related quadrilaterals on a square "screen" (namely as a
function of their diagonals) and by discussing their behavior according both to
the Grashof conditions and to the Regge symmetries, concertedly considering the
classification of the two mechanisms and that of the corresponding objects of
the quantum mechanical theory of angular momentum. An interesting topological
difference is demonstrated between mechanisms belonging to the two Regge
symmetric configurations: the movements in the Grashof cases span chirality
preserving configurations with a 2 pi-cycle of a rotating bar, while by
contrast the non-Grashof cases span both enantiomeric configurations with a 4
pi-cycle.
