The computation of the unoriented bordism group of the point $N_*=\Omega_*^O$ is a classic result.

I would like to know a related bordism group, where we specify the twisted fundamental class $[M]\in H_d(M,\mathbb{Z}^w)$ as part of the data. More precisely, I would like to consider the pairs $$ (M, [M]\in H_d(M,\mathbb{Z}^w)) $$ where $\mathbb{Z}^w$ is the coefficient system twisted by $w_1(M)$; I then call two $(M,[M])$ and $(M',[M'])$ bordant when there is $(N,[N]')$ such that $\partial N=M \sqcup M'$ where the twisted orientation of $N$ induces that of $M$ and $M'$, as in the case of oriented bordism.

Most probably this is well known to the experts...