We report a measurement of a new high spin Jπ=5− state at 22.4(2) MeV in C12 which fits very well to the predicted (ground state) rotational band of an oblate equilateral triangular spinning top with a D3h symmetry characterized by the sequence 0+, 2+, 3-, 4±, 5- with almost degenerate 4+ and 4- (parity doublet) states. Such a D3h 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 C12 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 C12 and light nuclei. In particular, the observation (or nonobservation) of the predicted (“missing”) states in the Hoyle band will allow us to conclude the geometrical arrangement of the three alpha particles composing the Hoyle state at 7.654 MeV in C12.