A theoretical model of a single molecule coupled to many vibronic modes is presented. At low energies, transport is dominated by electron-vibron processes where transfer of an electron through the dot is accompanied by the excitation or emission of quanta (vibrons). Because the frequency of the nth mode is taken as an nth multiple of the frequency of the fundamental mode, several energetically degenerate or quasidegenerate vibronic configurations can contribute to transport. We investigate the consequences of strong electron-vibron coupling in a fully symmetric setup. Several striking features are predicted. In particular, a gate asymmetry and pronounced negative differential conductance features are observed. We attribute these features to the presence of slow channels originating from the interplay of Franck-Condon suppression of transport channels with spin and/or orbital degeneracies.