  Controlling the quantum interface between two bosonic modes is essential in
countless implementations of quantum information processing. However, full
controllability is rarely achieved in most platforms due to specific physical
limitations. In this work, we completely characterize the linear two-mode
interfaces under the most pessimistic restriction that only single-mode
operation is available. When arbitrary Gaussian single-mode operations can be
applied to both modes, we find that every interface can be characterized by an
invariant transmission strength. Moreover, in the practical situation that
squeezing is restricted in one of the modes, we discover two additional
quantities, irreducible squeezing and irreducible shearing, that are invariant
under the allowable controls. By using this characterization, we develop
systematic strategies to engineer an arbitrary linear interface through
cascading multiple fixed component interfaces. Without squeezing restriction,
our protocol is optimal and requires at most three component interfaces. Under
the squeezing constraint, our protocol can be extended to engineer also the
additional invariants by using no more than two more rounds of cascade. We also
propose the remote squeezing scheme to tackle the squeezing restriction through
interfacing with an active auxiliary mode.
