For a generic dimension $d$, is there an nonorientable manifold $M$ (i.e. $w_1(TM)\neq 0$) with vanishing $w_1(TM)\cup w_1(TM)$ and $w_2(TM)$, i.e.,

$$w_1(TM)\cup w_1(TM)=0, ~~~~~ w_2(TM)=0, ~~~~~w_1(TM)\neq 0?$$

Here $w_i(TM)$ is the $i^{\text{th}}$ Stiefel-Whitney class of the tangent bundle of the manifold $M$. For $d=2$, the Klein bottle is an example.

If such manifolds exist, what kind of structure do they carry? For example, if $w_1(TM)=0$, and $w_2(TM)=0$, then the manifold can be equipped with a spin structure, and we say it is a spin manifold. I'd like to see what is the corresponding structure in the above more complicated case.