High-level ab initio calculations show that the singlet photochemical cis-trans isomerization of organic molecules under isolated conditions can occur according to two distinct mechanisms. These mechanisms are characterized by the different structures of the conical intersection funnels controlling photoproduct formation. In nonpolar (e.g. hydrocarbon) polyenes the lowest-lying funnel corresponds to a (CH)3 kink with both double and adjacent single bonds twisted, which may initiate hula-twist (HT) isomerization. On the other hand, in polar conjugated systems such as protonated Schiff bases (PSB) the funnel shows a structure with just one twisted double bond. The ground-state relaxation paths departing from the funnels indicate that the HT motion may take place in nonpolar conjugated systems but also that the single-bond twist may be turned back, whereas in free conjugated polar molecules such as PSB a one-bond flip mechanism dominates from the beginning. The available experimental evidence either supports these predictions or is at least consistent with them.