Was just reading about some of the Egyptian obelisks in Rome and how several were restored in the late 1500s by a pope after being found broken/toppled over. When I was in Rome several years ago I also remember the tour guide mentioning that the statues we saw of the Gemini Twins and their horses had been made during the Roman Empire era, but had broken apart and been restored/put back together during the Renaissance.
So..what is the deal? The statues/obelisks just toppled over and sat there in place for a thousand years? Was Rome totally abandoned? Was an earthquake involved? Did people steal smaller statues and take them home but these were too big to move? (Don't remember the Gemini twins being THAT big)
Toppling statues, or defacing them was common practice in the Roman world. It operated as an act of iconoclasm, and very often the treatment of statues mirrored the treatment of the bodies of those deposed. (See: Peter Stewart, "The Destruction of Statues in Late Antiquity", in Constructing Identities in Late Antiquity, Ed. Richard Miles, NY: Rutledge, 1999)
As for depictions of deities, this was often perpetrated as acts of Christian anti-pagan violence, or as Church sponsored iconoclasm. This, along with the repurposing of temple sites into Churches changed the "sacred" landscape of the late empire.