  We report a measurement of a new high spin Jp = 5- state at 22.4(0.2) MeV in
12C which fits very well to the predicted (ground state) rotational band of an
oblate equilateral triangular spinning top with a D_3h symmetry characterized
by the sequence 0+, 2+, 3-, 4+/-, 5- with almost degenerate 4+ and 4- (parity
doublet) states. Such a D_3h symmetry was observed in triatomic molecules and
it is observed here for the first time in nuclear physics. We discuss a
classification of other rotation-vibration bands in 12C such as the (0+) Hoyle
band and the (1-) bending mode band and suggest measurements in search of the
predicted ("missing") states that may shed new light on clustering in 12C and
light nuclei. In particular the observation (or non-observation) of the
predicted ("missing") states in the Hoyle band will allow us to conclude the
geometrical arrangement of the three alpha-particle composing the Hoyle state
at 7.654 MeV in 12C.
