The history of museums reveals paradigm shifts between the modern museum, a temple to the dominant culture and a factory of stereotypes marginalising minorities including people with disabilities, and the new museum, which challenges these assumptions and strives to welcome the whole population as its audience, to build a public sphere. Classicly, a museum's message is communicated in a linear fashion: from cultural production, through transmission, to the passive listening of a receiving public. We describe how cultural mediation gradually changes this pattern towards a circularity in which the public can become co-producers of discourse and representations, in a process of empowerment. By questioning the notion of universal accessibility using these communication models, we are led to evoke reverse mediation, as a transformation of the scheme that starts with an initiative from the minority, that allows a new look at the collection, and that comes to the service of the community. With an intersectional approach, we draw our examples from different situations involving different minorities and subcultures.