The coupling of transverse motion is a natural occurrence in
particle accelerators, either in the form of a residual coupling
arising from imperfections or originating by design from strong
systematic coupling fields. While the first can be treated
perturbatively, the latter requires a robust approach adapted to
strongly coupled optics, and a parametrization of the linear optics
must be performed to explore beam dynamics in such peculiar
lattices. This work highlights the key physical interpretations of
the main parametrization formalisms to describe linear coupled
optics, along with explicit links and comparisons of these
parametrizations. Concepts rarely illustrated in other works, such
as forced mode flips and local coupling, are explored in detail,
clarifying some anomalies that can arise in lattice functions. The
analytical methods have been implemented in a reference Python
package and connected with ray-tracing and integration codes to
explore examples of strongly coupled lattices, which are discussed
in detail to highlight the key physical interpretations of the
parametrizations and characteristics of the lattices.