There were countless rituals, sacrifices, festivals or ‘cults’ in Greco-Roman polytheism, and many different ways of classifying them. Some useful categories are defined in the ancient Latin dictionary of Sextus Pompeius Festus, On the Meaning of Words (De verborum significatu), generally under the umbrella term of sacra (‘sacred actions’). The most obvious, but by no means all-encompassing, distinction is between public and private rituals:
Public rituals (publica sacra) are those which are observed at public expense for the sake of the whole people, or for the mountains (of Rome), country districts (pagi), urban assemblies (curiae) and shrines (sacella).
Private rituals (privata) are those which are observed for the sake of individual persons, families and clans (gentes).
As the particular communal subdivisions of Rome and the families and clans no longer exist, the ‘extremes’, the most private and the most general, are now the most relevant:
Private festivals (privatae feriae) is the name of our personal rituals (sacra propria), like birthdays (dies natales), those relation to a private undertaking (operationis) and death rites (denecales).
Popular rituals are those, as Labeo says, which all citizens observe, and which are not attributed to specific families. E.g., Fornacalia [early February], Parilia [April 21st], Laralia [May 1st], porca praecidania [?].
Some (but not all) of the public festivals had fixed dates:
Fixed rites (stata sacrificia) are those which must take place on certain days.
Those with fixed and varying dates could be classed together in contrast to rituals performed for a specific occasion:
Customary rites (sollemnia sacra) are those which are used to take place at certain times or in certain years.
An example of a customary ritual that is not annual but only held “in certain years” would be the Secular Games, which were (in theory) held every century.
All these terms were defined for the Roman context, but they can easily transfered to others. Yet so far, the picture is very rigid: it appears that the individual is included in ever larger circles, up to the unit of a whole people, but little allowance is made for associations that cut across those circles.
We must be conscious of Festus’ context, however: writing in the 2nd century CE, he had an audience far beyond the city limits of Rome in mind, and as such, the “citizens” who observe the “popular rites” are not the Roman people in an ethnic, let alone racial sense. Rather, they are those across the empire who participate in the ritual culture of the rulers. This means colonies populated by emigrants from Rome, the military which recruited non-citizen subjects from across the Roman domain, people in cities that had adopted Latin dress and language, but also places that remained pointedly non-Roman while simultaneously being deeply entangled in the imperial world – like Athens: sought out by countless Roman travellers precisely because it was so paradigmatically Greek, but also home to a temple of Rome!
It is thus no surprise that there is a term in Festus’ work for those rituals independent of Roman history which continued to flourish under Rome’s dominance:
Provincial rituals (municipalia sacra) is the name for those which people have had from the beginning, before they received Roman citizenship, and which the pontifices have wished them to observe, and to do, in this manner, what they had been accustomed to from antiquity.
And on the other hand, the rituals at Rome were themselves not purely autochthonous:
Foreign rituals (peregrina sacra) is an appellation for those which were taken up either when gods were evoked* from cities at war with Rome, or which were requested in peace for the sake of certain religious obligations, as were those of Mater Magna from Phrygia, those of Ceres from Greece, and those of Aesculapius from Epidaurus. These are worshipped in the manner of those from whom they were derived.
(*Evocation is a ritual to persuade another city’s god to abandon it and favor Rome instead, to be worshipped there.)
Thus, Festus’ preoccupation with Rome is not ethnic in character (as if he favored ‘Roman polytheism’ over alternative religions); rather, it is an orientation towards the civic institutions of the ruling city, which as a matter of social reality have subordinated all other religious institutions to themselves.
The real weakness in Festus’ account is that he does not (in the extant portion of the text, at any rate) recognize ritual associations based on choice or interest, which are to a significant extent independent of familial and even geographic belonging. The closest thing is his entry on mysteries, but the focus here is on their secrecy, not who participates in them:
Secluded rituals (seclusa sacra) they used to call what the Greeks call the mysteries.