There were at least six (sub)genre of cachucha:

The 19th century minstrel Miss Lucy, in character, was a cross-dressing white man in black face doing the “hoochie-coochie,” for lack of a better term.

The only thing more “authentic” about the Lola Montez, Lotta Crabtree et al style of Spanish Dance was the female anatomy.

Both were taking off from La Cachucha, the romantic ballet folklorico popularized by Fanny Elssler in the 1830s. It was more “respectable,” to some, but it's still a modern dance, neither Andalusian nor Cuban afaik.

I got nothing for Cuba. The Andalusian form is a secular villancico adaptation of the Peruvian chachua.

The South American Roman Catholic music of the same name was a kind of learning tool and much older. Some of the first chachua mentions are kalenda holiday/carnival song. The villancico style today is reduced to Christmas elevator music.

Older still is the original Peruvian cachua, the only true folk dance of the lot.