The reaction of ethylene (Et) and its radical cation (Et.+) which has received much attention from the experimental side, was studied at the QCISD(T)/6-31G*//UMP2/6-31G* level of theory. According to these calculations the primary product of this reaction is a pi complex cation [Et...Et].+, where the two Et molecules are linked by a single bond of ~1.9 A and which is bound by 18.2 kcal/mol relative to Et + Et.+, in good agreement with experiment. Starting from this complex, two transition states were located, one leading to the radical cation of cyclobutane (Ea=9.0 kcal/mol) and the other involving a 1,3 H-shift leading to the radical cation of 1-butene (Ea=5.9 kcal/mol). No equilibrium structure corresponding to the frequently postulated tightly bound tetramethylene radical cation intermediate could be found, and therefore no such species seems to be involved in the well-documented rearrangements of [Et...Et].+ and/or ionized cyclobutane to butene radical cations. An intriguing discrepancy between the minimum-energy structures of the [Et...Et].+ on the UHF and UMP2 potential energy surfaces is discussed in terms of a bias introduced by the poor convergence of the Moller-Plesset perturbation series for species whose UHF wavefunction is strongly contaminated by higher spin states. A revised picture of the potential energy surfaces on which the various C4H4.+ rearrangements which were observed both in the gas phase and in condensed phase is proposed.