  Zamolodchikov's tetrahedron equations, which were derived by considering the
scattering of straight strings, can be written in three different labeling
schemes: one can use as labels the states of the vacua between the strings, the
states of the string segments, or the states of the particles at the
intersections of the strings. We give a detailed derivation of the three
corresponding tetrahedron equations and show also how the Frenkel-Moore
equations fits in as a {\em nonlocal} string labeling. We discuss then how an
analog of the Wu-Kadanoff duality can be defined between each pair of the above
three labeling schemes. It turns out that there are two cases, for which one
can simultaneously construct a duality between {\em all} three pairs of
labelings.
