  In this paper we test an approximate method that is often used in lattice
studies of the Landau gauge three-gluon vertex. The approximation consists in
describing the lattice correlator with tensor bases from the continuum theory.
With the help of vertex reconstruction, we show that this "continuum" approach
may lead, for general kinematics, to significant errors in vertex tensor
representations. Such errors are highly unwelcome, as they can lead to wrong
quantitative estimates for vertex form factors and related quantities of
interest, like the three-gluon running coupling. As a possible solution, we
demonstrate numerically and analytically that there exist special kinematic
configurations for which the vertex tensor structures can be described exactly
on the lattice. For these kinematics, the dimensionless tensor elements are
equal to the continuum ones, regardless of the details of the lattice
implementation. We ran our simulations for an $SU(2)$ gauge theory in two and
three spacetime dimensions, with Wilson and $\mathcal{O}(a^2)$ tree-level
improved gauge actions. Our results and conclusions can be straightforwardly
generalised to higher dimensions and, with some precautions, to other lattice
correlators, like the ghost-gluon, quark-gluon and four-gluon vertices.
