There were certainly events that made an impression on the Roman people, who wrote them down and carried them through history, so Livy has some true facts to use. One example was the barbarian sack of Rome in 390 BCE. That story was well-known to all Romans. A treaty between Rome and Carthage dated 507 BCE was also verified, as was a solar eclipse in 404 BCE.
We know that Rome was originally settled by two separate groups who tended goat and cattle herds, because of the differences in their pottery. They lived in huts on the top of two of the famous Roman hills: the Palatine and the Esquiline and lived a pastoral life, which continued for a century or more until the Etruscans appeared. More on that later.
Let us review the history of the kings in chronological order. Those stories that are bolded appear to be true.