A problem about Mazdakism is that, to my knowledge, we have relatively little reliable knowledge about it.
Generally speaking, contemporary written sources about the Sasanids are not abundant.
Historical narrative come largely from Islamic times. This means that they were written some centuries after the facts, and that they often reflect opinions, concerns and interests pretty far removed from what the average peasant under the Sasanid rulers felt or thought.
This is of course a general problem for pre-modern times, but it is particularly acute in pre-Islamic Iran. A LOT of contemporary records are either lost or poorly preserved.
I am not an expert on this particular topic, although I really wish I knew more (btw, I would be grateful to anyone who could point to accessible up-to-date scholarship), but I feel that a lot of the emphasis on "communistic" tendencies in Mazdakism might be source bias - of course, from the perspective of the people writing them, as a negative feature. Especially considering that the court seemingly did, for a time, support it, which sounds odd if communal property was really a core point (perhaps land reform was at stake? I don't know). If it is true that the movement had Gnostic influences (I've read conflicting claims on that), then Gnosis was generally hardly egalitarian.
Also, I am under the impression that it is difficult to define what is "inherent" to Mazdakism, since it was probably rather vague ideologically; anyway, it sounds like a particular interpretation of Mazdaic faith, not a specifically structured religion.