Let $X$ denote a $4$-manifold with boundary obtained by adding $k_1$ $1$-handles to $B^4$ and $k_2$ many $2$-handles to the resulting manifold i.e. $X$ is an arbitrary $4$-dimensional $2$-handlebody. Let $D(X)$ denotes its double. Evidently $D(X)$ is nullbordant as it is the boundary of $X\times [0,1]$.

My question is which nullbordant $4$-manifolds arise as doubles of $2$-handlebodies? If $M$ is an arbitrary closed $4$-manifold, can $M \# -M$ be written as such a double?