Why would you need the rope?
My understanding is that the victim is tied to the cross beam. He then carries the beam to Golgotha, is nailed or otherwise fastened to the beam, then the beam is hoisted up and over the upright, so that the upright slides into a hole in the cross beam. Thus the two pieces of the cross are locked together, and the feet are nailed into place.
Death by crucifixion occurs by suffocation. As Christ was suspended by His wrists (the nails were never put in the palms), His arms were stretched so that He could not exhale. Thus He was forced to push down on the nail in His feet, pushing Himself upwards and relieving the stretch. The awkward position of the feet (which were nailed with a single spike, and the soles of the feet flat against the upright) caused the legs to rapidly tire and cramp up. As His legs gave out, Christ would then sink slowly down until He was suspended by His wrists again. Repeat until death supervenes.
This is why they broke the legs of the two thieves on either side of Jesus. With their legs broken, they could no longer push themselves up to breathe, and would suffocate in a few minutes (if they had not already died of shock, exposure, blood loss, and dehydration). Jesus had already died by that time, as He was apparently in much worse shape before crucifixion. He had not slept since Wednesday, not eaten or drunk since Thursday and the Last Supper, and had been beaten up by the soldiers, flogged, and walked all over the city to Herod, and back again. No wonder He was unable to carry His cross to Golgotha, and had to be helped by Simon of Cyrene. Even so, Pilate was surprised that Jesus had died so quickly, when Joseph of Arimathea asked for the body.
Crucifying someone upside down would involve building a new upright, so that the cross beam could slide down far enough to allow the feet to be nailed in place over the head. Unless, as has been mentioned, it was an X-shaped cross, although I don’t know how that would be made to stand upright.
I have seen speculation that sometimes the cross used was a single upright. You would then lie the victim on the upright, nail his wrists to it, then place it upright and nail the feet in place. This might work for upside-down crucifixion, but it would be more trouble to nail both wrists and feet in place before putting the beam upright into the ground.
As I said, it is hard for me to believe that the soldiers would go to the trouble of anything besides SOP unless they were ordered to do so by their commanding officer. If the legend about Peter is true, he would have had to get his request for upside-down crucifixion approved by the condemning magistrate, who would then order the soldiers to give Peter a special crucifixion.
Regards,
Shodan