The Roman god associated with doors and beginnings, and the namesake of January, is Janus.
Janus was traditionally depicted with two faces, one looking forward and one looking backward. This imagery suited transitions, thresholds, entrances, departures, and the movement from one period to another. The Roman month Ianuarius, later January in English, was named for him.
New Year's symbolism often connects Janus with reflection on the past and anticipation of the future. That association is cultural and metaphorical; it does not mean that every New Year's Eve custom originated in ancient Rome.
Janus is sometimes confused with Saturn, another important Roman deity linked with a major winter festival, Saturnalia. Saturnalia influenced Roman seasonal celebrations but did not give January its name. Janus's distinctive role is as a god of beginnings, gates, and transitions.