topsy-turvy vs head over heels

topsy-turvy is an adjective but head over heels is not an adjective.

Word	Noun	Adjective	Verb	Adverb
topsy-turvy	No	Yes	No	Yes
head over heels	No	No	No	Yes

As adverbs, topsy-turvy and head over heels are synonyms defined as:

Other synonyms of topsy-turvy include heels over head, in great confusion, topsy-turvily.

topsy-turvy (adjective)	head over heels (adjective)
in utter disorder

topsy-turvy (adverb)	head over heels (adverb)
in disorderly haste	in disorderly haste
in a disordered manner