Founded on pagan traditions, January stems from Janus, the Roman deity with two faces who had the powers to look back (past) and to the future. The first day of January is in celebration of him.
He was the god of doorways and gates, endings and new beginnings, guarding the threshold, and was associated with powers of prophecy, an augur and mediator. At times he is connected to the sun and moon.
Janus was the ruler of Latium, and under his rule was responsible for the Golden Age that brought money, laws and agriculture to the area. It was a time of great peace and abundance.
From translation of Ovid Fast. 1.17 1-188 from Taylor's article:
"It was Roman tradition at the New Year to give the god honey, cakes, incense and wine to buy favorable signs and a guarantee of good luck. Gold brought better results than baser coins.
"Then I asked," Why, Janus, when I placate other gods, do I bring incense and wine to you first?"
"So that you may gain entry to whatsoever gods you wish," he replied, "through me, who guard the threshold."
"But why are glad words spoken on your Kalends? And why do we give and receive best wishes?"
Then the god, leaning on the staff in his right hand, said, "Omens are wont to reside in beginnings. You train your anxious ears on the first call, and the augur interprets the first bird he sees. The temples and ears of gods are open, no tongue intones wasted prayers, and words have weight." Janus had finished.
I was not silent for long, but tagged his final words with words of my own. "What do your dates and wrinkled figs mean, or the gift of honey in a snow-white jar?"
"The omen is the reason," said he - "so that the sweetness replicates events, and so that the year should be sweet, following the course of its beginnings."