I believe that there was multiple ways and that upside down wasn’t necessarily the norm

For example Seneca the Younger wrote as listed by the New World Encyclopedia (1)

Crucifixion was carried out in many ways under the Romans. Josephus describes multiple positions of crucifixion during the siege of Jerusalem in 70 C.E. when Titus crucified the rebels;[10] and Seneca the Younger recounts: “I see crosses there, not just of one kind but made in many different ways: some have their victims with head down to the ground; some impale their private parts; others stretch out their arms on the gibbet.”

At times the gibbet was only one vertical stake, called in Latin crux simplex or palus. This was the most basic available construction for crucifying. Frequently, however, there was a cross-piece attached either at the top to give the shape of a T (crux commissa) or just below the top, as in the form most familiar in Christian symbolism (crux immissa). Other forms were in the shape of the letters X and Y

1. https://www.newworldencyclopedia.org/entry/Crucifixion citation 2

Ive also listed the original source from their citation here

Dialogue “To Marcia on Consolation,” 6.20.3 The Latin Library. Retrieved February 21, 2019 with Google Translate link here