We investigate the dynamics of structured laser beams carrying angular momentum (AM) as they undergo conical diffraction in photonic graphene. Using a combination of numerical simulations and analytical calculations, we demonstrate a surprising result: the beam's orbital AM tracks the dynamics of the lattice pseudospin rather than the AM of its slowly varying envelope. This finding provides evidence that pseudospin behaves as a genuine form of angular momentum. Furthermore, we explore beam propagation in staggered honeycomb lattices governed by the massive Dirac equation, revealing that the orbital AM is highly sensitive to both the band gap size and the envelope's initial AM. We explain our numerical results by drawing an analogy with the orbital magnetization of electrons.