Janus is the Roman god whose name gives January its name and whose symbolism suits New Year's Eve's transition between years.
Janus was associated with beginnings, gates, doorways, transitions, and passages. Roman artistic representations commonly show him with two faces, traditionally interpreted as looking in opposite directions—one toward what has passed and one toward what is beginning.
The month January, Latin Ianuarius, was named for Janus. That association makes the deity especially relevant to New Year's customs, although many modern celebrations are not direct religious rites dedicated to him.
New Year's Day falls on January 1 in the Gregorian calendar used by much of the world, but calendars and New Year observances have varied across cultures and historical periods. The god's name is therefore a specific Roman connection, not evidence that every New Year's Eve tradition originated in ancient Rome.