We present a symmetry-based method for obtaining suitable tensor descriptions of lattice vertex functions without spinor components. The approach is based on finding the polynomial functions of vertex momenta, which satisfy the appropriate tensor transformation laws under hypercubic symmetry transformations. We use the method to find the most general possible (up to finite volume effects) basis decompositions for lattice vectors and second-rank tensors. The leading-order non-continuum versions of these representations are then applied to the Landau gauge gluon propagator and ghost-gluon vertex of Monte Carlo simulations, to reveal two interesting insights. First, it is demonstrated numerically and analytically that there exist special kinematic configurations where the basis descriptions of both functions reduce to their continuum analogues. Second, for the gluon two-point correlator it is shown that the rate at which the function approaches its continuum form in the infrared is independent of the lattice gauge coupling $\beta$ (when working in lattice units): the said rate depends on kinematics alone and is ultimately dictated by the numerical gauge-fixing procedure. We also comment on how this reflects on the lattice investigations of the anomalous magnetic moment of the muon. Finally, we argue how our findings can be used to directly test some of the continuum extrapolation methods.