Some exact results are given, that connect, for a general magnitude of the interaction term U, the repulsive and attractive Hubbard models, in terms of the eigenspectra and quantum diffusion properties. In particular, it is shown that, for some initial conditions, the quantum evolution cannot differentiate between the attractive and repulsive models. These results apply to both fermionic and bosonic models, in any dimension and for general filling, as far as the underlying structure is bi-partite.