Critical points and bifurcations are considered for the acetylene effective Hamiltonian in the polyad space of total bend and vibrational angular momentum quantum numbers [N-b, l]. A "phase diagram" is constructed for the surface of minimum energy critical points. The phases denote vibrational modes of different character, including new types of anharmonic modes born in bifurcations from the ordinary normal modes. A tetracritical point is the outstanding feature of the diagram. Patterns in the energy levels are considered. We start with a search for "defects" in the lattice of energy levels, similar to what is found with quantum monodromy in integrable systems. No such pattern is found. Instead, patterns are predicted in the first and second derivatives of the energy with respect to the quantum numbers, by close analogy to the theory of phase transitions. The tetracritical point finds striking manifestation in these patterns. This may be amenable to experimental test. Moreover, the minimum energy path, analogous to the reaction path, has derivative patterns that can already be compared with experiment. Agreement of theory and experiment is found within experimental error.