This a very sophisticated question. You are right that all things need to be interpreted. There is no such thing as objective truth. The more technical word for religious interpretation is called, exegesis. More importantly, exegesis occured among the Romans. Let me give you a few examples.
First, it is clear that there were many priesthood among the Romans - augurs, the quindecemviri (keepers of the Sibylline books), and the haruspices, to name a few. So, whenever the senate called upon the these priesthoods for religious advice, there was a fair amount of interpretation. The quindecemviri would look into the Sibylline books and advise a proper expiation or course of action. In a word, interpretation took place.
Second, The very fact that different expiations are advised for similar prodigies, which the Romans believed were from the gods, confirms the fact that exegesis of texts took place. In 436 B.C.E. the quindecemviri advised the senate to dedicate the temple of Apollo on account of pestilence (Liv. 4.21); in 399 B.C.E. the same priesthood advises the first lectisternium for pestilence (Liv. 4.25), and in 293 B.C.E. on account of more pestilence, the senate is advised to bring Aesculapius to Rome (Liv. 10.47).
From a methodological point of view, it is undeniable that interpretation was a central feature of Roman religion.