Let $E$ be a smooth real vector bundle over a smooth compact $n$-dimensional manifold $M$. It is relatively easy to see that the Stiefel-Whitney classes $w_i(E)$ are not diffeomorphism invariants of the total space of $E$.
We define the "Stiefel-Whitney numbers of $E$" in the obvious way. For $I=(i_1,i_2, \dots ,i_n)$ with $1i_1+2i_2+\dots+ni_n=n$ define $$W_I(E)=\langle w_{1}(E)^{i_1}w_{2}(E)^{i_2}\dots w_{n}(E)^{i_n},[M]\rangle\in \Bbb Z/2\Bbb Z$$
Are these $W_I(E)$ diffeomorphism (or homeomorphism) invariants of the total space of $E$? Specifically do any smooth vector bundles $E_1$ and $E_2$ over $M$ which have diffeomorphic total spaces have the same "Stiefel-Whitney numbers"?