We study C1 invariant, two-dimensional tori of constant energy without singularities of the Euler–Lagrange flow of a convex, superlinear Lagrangian defined in the tangent space of a closed surface. We show that in regular energy levels, such tori are graphs of the canonical projection if and only if they are minimizing. This result was known for C3 invariant, Lagrangian tori of Finsler Lagrangians, which implied the statement for C3 invariant Lagrangian tori of unimodal Lagrangians with a high-energy level. We also show that graphs have energy $E \geq c_0(L)$, where c0(L) is Mañé's strict critical value; and that the presence of Reeb components in such tori implies that the energy is c0(L). Moreover, if we assume that the Lagrangian is unimodal, we show that Lagrangian invariant tori with no singularities have no Reeb components, regardless of the energy level.