  We investigate the orthogonality preserving property for pairs of mappings on
inner product $C^*$-modules extending existing results for a single
orthogonality-preserving mapping. Guided by the point of view that the
$C^*$-valued inner product structure of a Hilbert $C^*$-module is determined
essentially by the module structure and by the orthogonality structure, pairs
of linear and local orthogonality-preserving mappings are investigated, not a
priori bounded. The intuition is that most often $C^*$-linearity and
boundedness can be derived from the settings under consideration. In
particular, we obtain that if $\mathscr{A}$ is a $C^{*}$-algebra and $T,
S:\mathscr{E}\longrightarrow \mathscr{F}$ are two bounded ${\mathscr A}$-linear
mappings between full Hilbert $\mathscr{A}$-modules, then $\langle x, y\rangle
= 0$ implies $\langle T(x), S(y)\rangle = 0$ for all $x, y\in \mathscr{E}$ if
and only if there exists an element $\gamma$ of the center $Z(M({\mathscr A}))$
of the multiplier algebra $M({\mathscr A})$ of ${\mathscr A}$ such that
$\langle T(x), S(y)\rangle = \gamma \langle x, y\rangle$ for all $x, y\in
\mathscr{E}$. In particular, for adjointable operators $S$ we have
$T=(S^*)^{-1}$, and any bounded invertible module operator $T$ may appear.
Varying the conditions on the mappings $T$ and $S$ we obtain further
affirmative results for local operators and for pairs of a bounded and of an
unbounded module operator with bounded inverse, among others. Also, unbounded
operators with disjoint ranges are considered. The proving techniques give new
insights.
