The knowledge of the intersection space topography of electronic states is essential for deciphering and predicting photoinduced reactions. Michl and Bonac̆ić-Koutecký developed a two-electron two-orbital model that allowed first systematic studies of the chemical origin of conical intersections in strongly polar systems. We generalize this approach to arbitrary functionalized and unfunctionalized polyene systems. For the extended model, a set of mathematical conditions for the formation of conical intersections are derived. These conditions are translated into geometrical motions and electronic effects, which help to explain and predict the structure and energetics of conical intersections. A three-step strategy for the conceptual search of conical intersections is outlined. Its universal validity is demonstrated using the textbook example cyclohexadiene and its functionalized derivative trifluoromethyl-indolylfulgide, a chromophore studied for possible application as a molecular switch.