El adverbio español tampoco corresponde a dos construcciones en inglés.

Afirmación Conformidad Contrario
I have never been to London. Me neither. (Yo tampoco.) Yes, I have. (Yo sí)
PERO: I have not been there either.
She has a dog. Me, too. (Yo también) / I have one, too. No, I have not. (Yo no)
She does not want to sleep. He neither. (Èl tampoco) Yes, I want. (Yo sí)
PERO: I do not want to sleep either.
They were always hungry. We, too. (Nosotros también)
We were, too.
No, we were not. (Nosotros no)