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The power of asteroseismology relies on the capability of global oscillationsto infer the stellar structure. For evolved stars, we benefit from uniqueinformation directly carried out by mixed modes that probe their radiativecores. This third article of the series devoted to mixed modes in red giantsfocuses on their coupling factors that remained largely unexploited up to now.  With the measurement of the coupling factors, we intend to give physicalconstraints on the regions surrounding the radiative core and thehydrogen-burning shell of subgiants and red giants. A new method for measuringthe coupling factor of mixed modes is set up. It is derived from the methodrecently implemented for measuring period spacings. It runs in an automated wayso that it can be applied to a large sample of stars.  Coupling factors of mixed modes were measured for thousands of red giants.They show specific variation with mass and evolutionary stage. Weak coupling isobserved for the most evolved stars on the red giant branch only; largecoupling factors are measured at the transition between subgiants and redgiants, as well as in the red clump.  The measurement of coupling factors in dipole mixed modes provides a newinsight into the inner interior structure of evolved stars. While the largefrequency separation and the asymptotic period spacings probe the envelope andthe core, respectively, the coupling factor is directly sensitive to theintermediate region in between and helps determining its extent.Observationally, the determination of the coupling factor is a prior to precisefits of the mixed-mode pattern, and can now be used to address furtherproperties of the mixed-mode pattern, as the signature of the buoyancy glitchesand the core rotation.

Period spacings in red giants; III. Coupling factors of mixed modes