The story
Pārvatī wanted Śiva and no one else. Her father had settled on Viṣṇu, so her friend carried her off to a forest cave — hariṇa, taken away, and ālikā, the friend, is one reading of the name Hartālikā. There she made a liṅga of sand and kept a fast without food or water until Śiva came and agreed to marry her.
The day is therefore about a woman choosing, and keeping the choice through a hard fast. Married women keep it for their husbands; unmarried women keep it for the husband they hope for.
In Karnataka the same tithi is Gauri Habba, and the frame is different: Gauri comes home to her parents for the day, and her son Gaṇeśa arrives the next morning to fetch her back. A house that installs Gaṇeśa on caturthī installs his mother the evening before.
