We develop an arbitrary-order locking-free method for linear elasticity on general (polyhedral, possibly nonconforming) meshes without nodal unknowns. The key idea is to reconstruct the relevant differential operators in terms of the (generalized) degrees of freedom by solving an inexpensive local problem inside each element. The symmetric gradient and the divergence operators are reconstructed separately. The divergence operator satisfies a commuting diagram property, yielding robustness in the quasi-incompressible limit. Locking-free error estimates are derived for the energy norm and for the L2-norm of the displacement, with optimal convergence rates for smooth solutions. The theoretical results are confirmed numerically, and the CPU cost is evaluated on both standard and general polygonal meshes.