A combination of quantum mechanics, semiclassical mechanics, and nonlinear classical dynamics is used to extract the detailed internal molecular motions that underly the quantum eigenstates of acetylene with 16 quanta of total bend excitation. No potential energy surface is used; rather, the states are represented by an algebraic effective Hamiltonian that has been extensively refined against experimental data. The classical mechanical analysis reveals widespread chaos, but the quantum mechanical structure is surprisingly regular. Specifically, all 81 quantum states can be assigned a pair of semiclassical quantum numbers that reveal the underlying classical motions associated with each state. These classical motions range continuously between limiting-case motions that we refer to as local bend (one hydrogen bending) and counter-rotation (the two hydrogens undergoing circular motions in planes perpendicular to the CC axis). The first reason that the regularity in the quantum structure was previously undetected is that the identification of regular nodal coordinates, if any exist, of quantum wave functions in a multidimensional (i.e., greater than two dimensions) space is generally a difficult task; our success here was made possible by the identification in a reduced two-dimensional (2D) space of two families of periodic orbits (dynamic modes) which evolve with energy. Every quantum state reflects the quantization of the two dynamic mode system. The second reason for the undetected regularity is that the regular sequences of quantum levels that we have identified are interspersed among each other in energy, thus giving the appearance of a complex, unassignable spectrum.