Odr (pronounced “Oh-der,” from Old Norse Óðr, “ecstasy, inspiration, fury, frenzy;” sometimes shortened to Óð or “Od”) is an obscure, seldom-mentioned god. According to the medieval Icelandic scholar Snorri Sturluson, Odr is the husband of the Vanir goddess Freya, who is the mother of his daughter Hnoss.
Snorri also briefly mentions one story about Odr, in which the god plays an entirely passive role: once, Odr went far away from the other deities. His destination and the reasons for his departure are never stated. Freya searched in vain for him, and wept tears of gold in his absence.
Odr’s existence, at least, is corroborated by two of the poems in the Poetic Edda, and he is alluded to in an 11th-century poem by Einarr Skulason. It’s therefore impossible for him to have been merely an invention of Snorri’s; Odr was an authentic, if perhaps rather late, feature of pre-Christian Norse mythology.
As sparse and cryptic as these mentions of Odr in Old Norse literature are, the evidence points ineluctably toward a single interpretation of Odr: he was an only nominally distinct counterpart to Odin.
Freya, Odr’s wife, can hardly be distinguished from Odin’s wife, Frigg, as I show in the articles on Frigg and Freya. The name Odin (Old Norse Óðinn) is Óðr with the masculine definite article (-inn) attached to the end to mean “master of óðr” or “exemplar of óðr.” Odin’s and Odr’s names, therefore, are practically identical. Odin was once exiled by the other gods for a long period, and Odr’s absence surely corresponds to this time.
However, it’s noteworthy that Snorri doesn’t mention any children of Odr besides Hnoss. Odin has numerous children, including very prominent gods such as Thor and Baldr. If Snorri had seen Odr and Odin as being truly identical, he certainly would have mentioned Odin’s children amongst Odr’s.
It seems that by the medieval period at the latest, and quite possibly in the earlier Viking Age, Odin had been split into two gods, and Freya/Frigg had been split into two goddesses. This change must have occurred relatively shortly before the 11th century, given how indistinguishable the characters of the bifurcated deities still were to each other by the time the major Old Norse literary sources were written.
This, of course, begs the question of why this change occurred. While we aren’t certain of the reasons why, it’s probably highly significant that in each of the split pairs, one deity belonged to the Aesir tribe of gods, and the other to the Vanir tribe of gods. The split between the Aesir and the Vanir was itself unique to later Norse mythology; such a division of the deities doesn’t seem to have occurred amongst any of the other branches of the Germanic peoples, nor amongst the Norse in earlier times. And as with the two pairs of deities we’ve been considering here, it’s difficult to point to any concrete features that distinguish the Aesir and the Vanir. There may have been differing tendencies or differences of emphasis, but any formulation that states that “the Aesir were the gods of such-and-such” and “the Vanir were the gods of these other things” is oversimplifying.
Perhaps, then, the split between Odr and Odin and between Freya and Frigg occurred as part of the larger split between the Vanir and the Aesir. This division would have still been young enough at the time of Christianization to be more of a change of name than of substance, given how difficult to differentiate the characters of the deities in question remained.
That suggestion, in turn, begs the question of why the gods were being cut into two groups just prior to the eleventh century. Unfortunately, however, no one knows the answer to that question.
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 Snorri Sturluson. The Prose Edda. Gylfaginning 34, Skáldskaparmál 20, 35.
 The Poetic Edda. Völuspá, stanza 25.
 The Poetic Edda. Hyndluljóð, stanzas 46-47.
 Simek, Rudolf. 1993. Dictionary of Northern Mythology. Translated by Angela Hall. p. 249-250.