We aim to explore the capabilities of dust emission and rays for probing the properties of the interstellar medium in the nearby anti-centre region, using gamma-ray observations with the Fermi Large Area Telescope (LAT), and the thermal dust optical depth inferred from Planck and IRAS observations. In particular, we aim at quantifying potential variations in cosmic-ray density and dust properties per gas nucleon across the different gas phases and different clouds, and at measuring the CO-to-H2 conversion factor, X$_{CO}$ , in different environments. We have separated six nearby anti-centre clouds that are coherent in velocities and distances, from the Galactic-disc background in HI 21-cm and $^{12}$CO 2.6-mm line emission. We have jointly modelled the gamma-ray intensity recorded between 0.4 and 100 GeV, and the dust optical depth at 353 GHz as a combination of HI-bright, CO-bright, and ionised gas components. The complementary information from dust emission and gamma rays was used to reveal the gas not seen, or poorly traced, by HI , free-free, and $^{12}$CO emissions, namely (i) the opaque HI and diffuse H$_2$ present in the Dark Neutral Medium at the atomic-molecular transition, and (ii) the dense H$_2$ to be added where $^{12}$CO lines saturate. The measured interstellar gamma-ray spectra support a uniform penetration of the cosmic rays with energies above a few GeV through the clouds. We find a gradual increase in grain opacity as the gas becomes more dense. The increase reaches a factor of four to six in the cold molecular regions that are well shielded from stellar radiation. Consequently, the X$_{CO}$ factor derived from dust is systematically larger by 30% to 130% than the gamma-ray estimate. We also evaluate the average gamma-ray X$_{CO}$ factorfor each cloud, and find that X$_{CO}$ tends to decrease from diffuse to more compact molecular clouds, as expected from theory.
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