The Catholic Encyclopedia tells us that the Romans gave gifts on 1st January (the Kalends of January), called strenae.
Pagan customs centering round the January calends gravitated to Christmas. Tiele (Yule and Christmas, London, 1899) has collected many interesting examples. The strenae (étrennes) of the Roman 1 January (bitterly condemned by Tertullian, de Idol., xiv and x, and by Maximus of Turin, Hom. ciii, de Kal. gentil., in P.L., LVII, 492, etc.) survive as Christmas presents, cards, boxes.
By us, to whom Sabbaths are strange, and the new moons and festivals formerly beloved by God, the Saturnalia and New-year’s and Midwinter’s festivals and Matronalia are frequented — presents come and go — New-year’s gifts (strenae) — games join their noise-banquets join their din! (ch. 14) … New-year’s gifts likewise must be caught at, and the Septimontium kept (ch. 10).
A google search reveals that “Tiele” is Tille, and on Google books here. I will have to read this, as it seems copiously referenced. Not sure whether the text is quite sensible, but it does contain interesting snippets.
But I can see at once, on p.84 n.3, a reference to Plautus, Stichus, iii. 2, 6; v. 2. 24; Ovid, Fasti, i. 187; Martial, viii.33, xiii.37; Seneca, Letters, 87. There are two unreferenced claims; that money took the place of New Year’s gifts under Augustus, and that the custom persisted to the time of Honorius and Arcadius.
There is a reference to the Kalends and the celebration of Janus in the Acts of the Council of Turin in 567 AD. (p.87 n.1), which calls him a king, not a god. In the capitula of Martin of Braga, chapter 73, we read:
Non liceat iniquas observationes agere Kalendarum, et otiis vacare gentilibus, neque lauro aut viridate arborum cingere domos.
Hanging up green boughs seems to be the custom. It would be interesting to know more about this.