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Oscillation modes with a mixed character, as observed in evolved low-massstars, are highly sensitive to the physical properties of the innermostregions. Measuring their properties is therefore extremely important to probethe core, but requires some care, due to the complexity of the mixed-modepattern. This work aims at providing a consistent description of the mixed-modepattern of low-mass stars, based on the asymptotic expansion. We also aim atstudying the variation of the gravity offset $\varepsilon_{g}$ with stellarevolution. We revisit previous work about mixed modes in red giants andempirically test how period spacings, rotational splittings, mixed-mode widthsand heights can be estimated in a consistent view, based on the properties ofthe mode inertia ratios. From the asymptotic fit of the mixed-mode pattern of alarge set of red giants at various evolutionary stages, we derive unbiased andprecise asymptotic parameters. As the asymptotic expansion of gravity modes isverified with a precision close to the frequency resolution for stars on thered giant branch (10$^{-4}$ in relative values), we can derive accurate valuesof the asymptotic parameters. We decipher the complex pattern in a rapidlyrotating star, and explain how asymmetrical splittings can be inferred, as wellas the stellar inclinations. This allows us to revisit the stellar inclinationsin two open clusters, NGC 6819 and NGC 6791: our results show that the stellarinclinations in these clusters do not have privileged orientation in the sky.The variation of the asymptotic gravity offset along with stellar evolution isinvestigated in detail. We also derive generic properties that explain underwhich conditions mixed modes can be observed.

Period spacings in red giants IV: Toward a complete description of the mixed-mode pattern