  The elements in the hyperoctahedral group $\mathfrak{B}_n$ can be treated as
signed permutations with the natural order $\cdots<-2<-1<0<1<2<\cdots$, or as
colored permutations with the $r$-order
$-1<_r-2<_r\cdots<_r0<_r1<_r2<_r\cdots$. For any $\pi\in\mathfrak{B}_n$, let
$\operatorname{des}^B(\pi)$ and $\operatorname{ides}^B(\pi)$ be the number of
descents and inverse descents in $\pi$ under the natural order, and let
$\operatorname{des}_B(\pi)$ and $\operatorname{ides}_B(\pi)$ be the number of
descents and inverse descents in $\pi$ under the $r$-order. In this paper, by
investigating signed permutation grids under both the natural order and the
$r$-order, we give combinatorial proofs for six recurrence formulas of the
joint distribution of descents and inverse descents over the hyperoctahedral
group $\mathfrak{B}_n$, the set in involutions of $\mathfrak{B}_n$ denoted by
$\mathcal{I}_n^B$, and the set of fixed-point free involutions in
$\mathfrak{B}_n$ denoted by $\mathcal{J}_n^B$, respectively. Some of these six
formulas are new, and some reveal the combinatorial essences of the results
obtained by Visontai, Moustakas and Cao-Liu through algebraic approaches such
as quasisymmetric functions. Furthermore, from these formulas, we conclude that
$(\operatorname{des}^B,\operatorname{ides}^B)$ and
$(\operatorname{des}_B,\operatorname{ides}_B)$ are equidistributed over both
$\mathfrak{B}_n$ and $\mathcal{I}_n^B$, but not on $\mathcal{J}_n^B$.
