  The classical part of the QCD partition function (the integrand) has,
ignoring irrelevant exact zero modes of the Dirac operator, a local SU(2N_F)
\supset SU(N_F)_L \times SU(N_F)_R \times U(1)_A symmetry which is absent at
the Lagrangian level. This symmetry is broken anomalously and spontaneously.
Effects of spontaneous breaking of chiral symmetry are contained in the
near-zero modes of the Dirac operator. If physics of anomaly is also encoded in
the same near-zero modes, then their truncation on the lattice should recover a
hidden classical SU(2N_F) symmetry in correlators and spectra. This naturally
explains observation on the lattice of a large degeneracy of hadrons, that is
higher than the SU(N_F)_L \times SU(N_F)_R \times U(1)_A chiral symmetry, upon
elimination by hands of the lowest-lying modes of the Dirac operator. We also
discuss an implication of this symmetry for the high temperature QCD.
