Is any symmetry element conserved during a supra-antarafacial cyclo addition of two cis-butadiene units

I tried to construct a correlation diagram for the [4+4] cyclo addition of two cis-butadiene units. But I failed to construct any meaning full orbitals for the “educt” side where the two butadiene molecules still have their pi-systems intact.

I thought that a C2 rotation axis is conserved during a supra-antarafacial reaction pathway but aligning the pi system of both units leads to a conflict. The backbone of the supra-facial part cannot be combined with the antarafacial unit to sustain C2 symmetry. This would only be possible for a combination of a cis-butadiene with a trans-butadiene. But without any conserved symmetry element correlation is impossible.

So I’m wondering if the Woodward-Hoffmann rules are applicable at all in this case or if I am being too strict with symmetry. Is it legit to neglect the backbone when thinking about the symmetry?