The Roman god traditionally associated with looking backward and forward at New Year's is Janus.
Janus was the Roman deity of beginnings, transitions, passages, doorways, and endings. He is commonly depicted with two faces, one looking toward the past and the other toward the future. That imagery made him a natural symbol for the turning of the year, even though many modern New Year's Eve customs developed much later.
The month of January takes its English name from Janus through the Latin Ianuarius. In the Roman calendar reforms associated with Julius Caesar, January became the first month of the year, helping strengthen the god's connection with annual beginnings. Roman religious practice also included the festival of the Kalends of January.
Janus is not the same figure as Saturn, whose festival Saturnalia is often mistakenly treated as the direct origin of modern New Year's Eve celebrations. New Year's traditions have many different sources, including Roman calendrical practice, Christian observances, local folklore, and modern civic countdowns.