During the Roman Republic, during the times of Cicero (in the first century BCE), one could visit the temple of Jupiter Stator, which according to the Romans was founded directly by Romulus himself, the first king of Rome.
Moreover, at the Forum Romanum there was a tree (exaggerated) under which the brothers were thrown across the river; in turn, on the Palatine, there was a thatched cottage in which they lived and grew up. However, as one traveler from the 1st century BCE says:
They don’t add anything to it that would make it more dignified, but if any part of it is destroyed, by bad weather or time, they fix it and restore it, if possible, to its previous state1.
As you can see, it wasn’t a real hut from the legend. Naturally, archaeologists did not come across any traces of this construction – due to the fragility of the material. Interestingly, however, the said cottage existed until the 4th century CE, probably still being rebuilt, because it was listed on the list of noteworthy buildings in Rome.
As you can see, even the Romans needed symbols and artifacts from ancient times to strengthen their attachment to their homeland and keep tradition alive.