at odds vs contradictory

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

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

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

Other synonyms of at odds include conflicting, self-contradictory.

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

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