While considerable progress has been made in developing promising subsystems for muon beam cooling channels to provide the extraordinary reduction of emittances, there is no end-to-end design that is capable of matching between or within the various subsystems. We present concepts to match emittances between and within muon beam cooling subsystems via the Helical Cooling Channel (HCC), which allows a general analytic approach to guide designs of transitions from one set of cooling channel parameters to another. These principles are demonstrated between segments in an existing cooling channel design, resulting in better performance (elimination of particle losses and colder muons) achieved in a channel approximately half its original length! These techniques will allow for a design of a complete cooling channel in a Muon Collider (MC) applicable to a Higgs Factory and an Energy Frontier machine.