Heat capacity (C), magnetic torque, and proton NMR relaxation rate (1/T(1)) measurements were performed on Fe6:Li single crystals in order to study the crossings between S = 0 and S = 1 and between S = 1 and S = 2 magnetic states of the molecular rings, at magnetic fields B(c1) = 11.7 T and B(c2) = 22.4 T, respectively. C vs B data at 0.78 K show that the energy gap between two states remains finite at B(c)'s (Delta(1)/k(B) = 0.86 K and Delta(2)/k(B) = 2.36 K) thus proving that levels repel each other. The large Delta(1) value may also explain the anomalously large width of the peak in 1/T(1) vs B, around B(c1). This anticrossing, unexpected in a centrosymmetric system, requires a revision of the Hamiltonian.