  For each adjointable idempotent $Q$ on a Hilbert $C^*$-module $H$, a specific
projection $m(Q)$ called the matched projection of $Q$ was introduced recently
due to the characterization of the minimum value among all the distances from
projections to $Q$. Inspired by the relationship between $m(Q)$ and $Q$,
another term called the quasi-projection pair $(P,Q)$ was also introduced
recently, where $P$ is a projection on $H$ satisfying $Q^*=(2P-I)Q(2P-I)$, in
which $Q^*$ is the adjoint operator of the idempotent $Q$ and $I$ is the
identity operator on $H$. Some fundamental issues on quasi-projection pairs,
such as the block matrix representations for quasi-projection pairs and the
$C^*$-morphisms associated with quasi-projection pairs, are worthwhile to be
investigated. This paper aims to make some detailed preparations. Two objects
called the semi-harmonious quasi-projection pair and the harmonious
quasi-projection pair are introduced and are systematically studied in the
general setting of the adjointable operators on Hilbert $C^*$-modules. Some
applications concerning the common similarity of operators and a norm equation
associated with the Friedrichs angle are also dealt with. Furthermore, many
examples are provided to illustrate the non-triviality of the associated
characterizations.
