At the start of things Brahma peopled the world, but everything he made was male and nothing he made could make more. He went to Shiva for help. Shiva answered by showing himself as a single body in two halves, the right his own and the left the goddess. From her side she released the female energies that would from then on carry creation forward, and Brahma's work could go on. The tradition remembers the moment in the image of Ardhanarishvara, the Lord who is half woman, and reads it as a statement that the two principles were never separate. The oldest surviving carvings of the form come from Kushan Mathura in the first century.
The Shakta Kalika Purana turns the story on a misunderstanding. Parvati saw a woman's face in the polished surface of Shiva's chest and took it for a rival; the reflection was her own. When the quarrel was settled she asked that she might never again be parted from him, but live in his very body, and Shiva took her as his left half.
What is different here:
Parvati's jealousy of her own reflection is the trigger; the union is her request for permanence, not a device for creation.
The Vamana Purana links the form to the demon Andhaka. When the blind demon who had lusted after the goddess was pierced on Shiva's trident and at last repented, he was received among Shiva's attendants as the gana Bhringi, and the same one-sided devotion that the Tamil story tells follows from there.
What is different here:
Supplies a back-story for Bhringi rather than a new origin: the devotee is the redeemed Andhaka.
In the Tamil telling the cause is a stubborn devotee. The sage Bhringi would honour Shiva alone: when he circled the Lord he refused to walk around Parvati beside him, and when the two sat together he turned into a bee to fly between them. Stung, the goddess undertook austerities and asked for a boon that would make his refusal impossible: to be joined with Shiva in one body. So the left half became hers, and Bhringi, still trying to circle only the male side, is remembered as the reason the form exists. The hill temple at Tiruchengode is dedicated to this form and takes the image out on Karthigai Deepam.
What is different here:
The cause is a devotee's partiality and Parvati's boon, not Brahma's stalled creation; the goddess is the mover of the story.
At Tiruvannamalai the temple tells that Parvati once covered Shiva's eyes in play and the worlds went dark. To make amends she came to the red hill and did penance there, and on the full moon of Karthigai Shiva took her back into his own body as his left half. The temple keeps the day as its festival of reunion, and the hill beacon is lit that night.
What is different here:
Parvati's penance at Arunachala after blinding Shiva in play; tied to Karthigai Deepam rather than to creation or to Bhringi.
Sources: Ardhanarishvara — Wikipedia; The Siva Purana (4 vols), tr. J. L. Shastri — via Wisdom Library; Andhaka — Wikipedia; Arunachalesvara Temple — Wikipedia