In spite of their extensive conjugation, fullerenes behave chemically and physically as electron-deficient alkenes rather than electron-rich aromatic systems. A qualitative molecular-orbital treatment and group-theoretical analysis accounts for this by showing that C-60 and all other closed-shell fullerenes C-n (n = 60 + 6k (k not equal 1)) of the infinite leapfrog class have six low-lying empty orbitals matching the translational and rotational symmetries. Six acceptor levels are also found for the other main class of closed shells, C-70 and the carbon cylinders C-n (n = 70 + 30k, n = 84 + 36k, all k).