The elusive connection between dynamics and local structure in supercooled liquids is an important piece of the puzzle in the unsolved problem of the glass transition. The Johari-Goldstein β relaxation, ubiquitous in glass-forming liquids, exhibits mean properties that are strongly correlated to the long-time α dynamics. However, the former comprises simpler, more localized motion, and thus has perhaps a more straightforward connection to structure. Molecular dynamics simulations were carried out on a two-dimensional, rigid diatomic molecule (the simplest structure exhibiting a distinct β process) to assess the role of the local liquid structure on both the Johari-Goldstein and the relaxation. Although the average properties for these two relaxations are correlated, there is no connection between the and properties of a given (single) molecule. The propensity for motion at long times is independent of the rate or strength of a molecule's β relaxation. The mobility of a molecule averaged over many initial energies, a measure of the influence of structure, was found to be heterogeneous, with clustering at both the and timescales. This heterogeneity is less extended spatially for the than for the dynamics, as expected; however, the local structure is the more dominant control parameter for the process. In the glassy state, the arrangement of neighboring molecules determines entirely the relaxation properties, with no discernible effect from the particle momenta.