contradictory vs at odds

contradictory is a noun but at odds is not a noun.

Word	Noun	Adjective	Verb	Adverb
contradictory	Yes	Yes	No	No
at odds	No	Yes	No	No

As adjectives, contradictory and at odds are synonyms defined as:

Other synonyms of contradictory include conflicting, self-contradictory.

contradictory (noun)	at odds (noun)
two propositions are contradictories if both cannot be true (or both cannot be false) at the same time

contradictory (adjective)	at odds (adjective)
of words or propositions so related that both cannot be true and both cannot be false	in disagreement
unable for both to exist or be true at the same time
in disagreement
that confounds or contradicts or confuses