Currently CLDR supports several types of cyclicNameSets, used primarily for East Asian luni-solar calendars:
years | months | days | dayParts | zodiacs
These can support format/standalone and various lengths:
Another type of cyclic name set important for East Asian luni-solar calendars is the set of 24 solar terms (in pairs of minor/major), which are tied to points in the solar cycle and thus approximately to specific dates in e.g. the Gregorian calendar.
I would like to add these as another cyclicNameSet type, "solarTerms". Coverage would be 80/modern for Chinese, Japanese, Korean, Vietnamese and probably 100/comprehensive for everything else. I can actually populate those languages from Wikipedia data, see http://en.wikipedia.org/wiki/Solar_term
One question is what to put in root, the English description or the pinyin?
Added this, populated for zh, zh_Hant, ja, ko, vi (the latter was missing other cyclicNameSet data, added it)