Not sure if this was ever mentioned on this forum before, but there seems to be a lot of evidence to support the fact that the Sun Moon Sect in XAJH is a trickle-down of the Ming Sect of HSDS.
For one, if you put the Sun and Moon characters together, you get the Ming character. After leadership of the Ming Sect was given to Zhu Yuanzhang by Zhang Wuji, he organized and lead the revolt that overthrew the Yuan government. He then ascended the throne himself and formed the Ming Dynasty, yes JY made the implication that the Ming Dynasty got its name from the Ming Sect. Anyway, my theory is that what most likely occurred is that Zhu turned power hungry and paranoid about his former Ming Sect buddies and most likely purged and hunted them down. Former Ming Sect members probably changed the name and organized under Sun Moon Sect, as a figurative way of showing the split. Since they were forced to hide from the Imperial Government of the Ming, the Sun Moon Sect most likely hide to stick to shady business and eventually turned into the evil-doing sect that it was in XAJH.
Other notable similarities: Ming Sect and Sun Moon Sect both have the same titles for their "Grand Left and Grand Right Hand Men". Yang Xiao and Fan Yao in HSDS, Xiang Wentian and Qu Yang in SoD. Although the Sun Moon Sect seemed to have shed all the religious rituals and rites that the old Ming Sect held, they still did seem to carry a certain mysticism that seemed reminiscent of the Ming.