Figure 4 shows the orbital array involved in the butadiene to cyclobutene interconversion. It is seen that there are four orbitals in this cyclic array. Thus in the interconversion reactions orbitals 1 and 4 overlap either in a conrotatory or a disrotatory fashion. Also, it is seen that the conrotation involves one plus-minus overlap as drawn while the disrotation involves zero plus-minus overlaps as drawn. Thus the conrotation uses a Möbius array while the disrotation uses a Hückel array.
But it is important to note, as described for the generalized orbital array in Figure 3, that the assignment of the basis-set p-orbitals is arbitrary. Were one p-orbital in either reaction mode to be written upside-down, this would change the number of sign inversions by two and not change the evenness or oddness of the orbital array. 
With a conrotation giving a Möbius system, with butadiene's four electrons, we find an "allowed" reaction model. With disrotation giving a Hückel system, with the four electrons, we find a "forbidden" reaction model.
Although in these two examples symmetry is present, symmetry is not required or involved in determination of reaction allowedness versus forbiddenness. Hence a very large number of organic reactions can be understood. Even where symmetry is present, the Möbius–Hückel analysis proves simple to employ.