A classical microscopic theory of rovibrational motion at high angular momenta in symmetrical nonlinear molecules AB, is derived within the framework of small oscillations near the stationary states of a rotating molecule. The full-dimensional analysis including stretching vibrations has confirmed the existence of the bifurcation predicted previously by means of the rigid-bender model [see B. I. Zhilinskii and I. M. Pavlichenkov, Opt. Spectrosk. (USSR) 64, 413 (1988)]. The formation of fourfold energy clusters resulting from the bifurcation has been experimentally verified for H2Se and it has been demonstrated in fully-dimensional quantum mechanical calculations carried out with the MORBID computer program. We show in the present work that apart from the level clustering, the bifurcation produces physically important effects including molecular symmetry-breaking and a transition from the normal mode to the local mode limit for the stretching vibrations due to the centrifugal forces. The application of the present theory with realistic molecular potentials to the H2Te, H2Se, and H2S hydrides results in predictions of the bifurcation points very close to those calculated previously. However for the lighter H2O molecule we find that the bifurcation occurs at higher values of the total angular momentum than obtained in previous estimations. The present work shows it to be very unlikely that the bifurcation in H2O will lead to clustering of energy levels. This result is in agreement with recent variational calculations.