Singular or plural in “between the 17ᵗʰ and the early 20ᵗʰ century/centuries”?

Which is correct in the following sentence, century or centuries?

[. . .] courtship gifts common in Sweden and other Scandinavian countries between the 17ᵗʰ and early 20ᵗʰ century.

Centuries is correct, because you're simultaneously referring to more than one century as the endpoints of a range. You could also say: "between the 17th century and the early 20th century" because in that case you're referring to each century individually.

I don't know about "correct", but both sound okay, to me. It's possible to make a grammatical argument for singular "century" in your title: "between the 17th and the early 20th century". There are two "the"s, so there are two noun phrases in the phrase, each of which is logically singular, since it refers to a single century. (The instance of "century" in the first noun phrase is elided by a process called "right node-raising".)

However, in the quote given in your question you've put just a single "the", so here there is only one noun phrase, whose head noun should be plural, since it refers to two centuries.

According to COS:

"the eighteenth and early nineteenth centuries"/ "from the twentieth to the twenty-first century"/"eighteenth- and nineteenth-century technologies"