We discuss the symmetry properties of a system of two (identical) interacting bosonic wells occupied by N bosons based on the Schwingerrealization of spins of the two-well Hamiltonian H, and show how such symmetries influence the energy spectrum form. We formulate the corresponding eigenvalue equation in a recursive N-dependent form that makes evident the mechanism causing the occurrence of the spectrum level doublets. Such doublets allows us to recover, in the classical limit, the ymmetry-breaking effect that characterizes the system classically. Special states solving the two-well Schroedinger problem are also presented which seems to reproduce at the quantum level the classical behavior of the population imbalance variable.