[cygx]

The terms co- and contravariant make sense on a purely algebraic basis, with components of tensors transforming 'the same as' or 'opposite to' the basis vectors. That the basis transformation is induced by transformations of some base manifold is incidental.

[ummonk]

Exactly. The fact that the bases are related to coordinates on the manifold is a property of differential geometry but the laws for transformation between bases are more general.