Lewis & Short

Ăgōnālĭa, ĭum or ōrum (like Saturnalia, Parentalia, etc.), n., a festival in honor of Janus, celebrated in Rome on the 9th of Jan. (V. Id. Jan.) and 21st of May (XII. Kal. Jun.).
Different derivations of the word were given by the ancients, concerning which see Ov. F. 1, 319-332. Ovid, in l. c., derives it from agonia, q. v. For other etym., v. the foll. art., and under agonium.