We employ a new laterally coupled, vertical double dot with a tunable tunnel-coupling gate in a parallel configuration to study the electron spin and orbital dependence of quantum mechanical tunnel coupling on the size of the honeycomb vertices in the small electron numbers regime. We find a transition from the weak coupling regime, where fluctuations in tunnel coupling due to varying electron configuration dominate the anticrossings, to a regime where the two dots coalesce. We apply a magnetic field to ascertain the orbital angular momenta of the Fermi surface eigenstates, which correlate with anticrossing size, and we identify spin pairs with congruent behavior.