We prove that if the Walsh bipartite map $ℳ=𝒲\left(\mathscr{H}\right)$ of a regular oriented hypermap $\mathscr{H}$ is also orientably regular then both $ℳ$ and $\mathscr{H}$ have the same chirality group, the covering core of $ℳ$ (the smallest regular map covering $ℳ$) is the Walsh bipartite map of the covering core of $\mathscr{H}$ and the closure cover of $ℳ$ (the greatest regular map covered by $ℳ$) is the Walsh bipartite map of the closure cover of $\mathscr{H}$. We apply these results to the family of toroidal chiral hypermaps ${(3,3,3)}_{b,c}={𝒲}^{-1}{\{6,3\}}_{b,c}$ induced by the family of toroidal bipartite maps...