Common meaning:

A person who is aware of their own motivations and can guess the motivations of others.

Hidden meaning of the Equity religion: 

All members of minoritized groups have unique insight into the nature of society. They know both their own truth and the truth of oppressors.

Discussion:

The Equityists declare that only they have the real truth, or to put it more philosophically, they are the only ones with gnostic knowledge of reality. They both know the truth of the oppressed and the truth of the oppressors.

According to Equity believers, the way double consciousness works is that  people in the oppressor group have no consciousness while those in the oppressed group have double consciousness. Members of the dominant group have no ability to self-reflect and understand what they are doing, while members of the minoritized group have the ability to both reflect on what they have experienced and to know the reality of the dominant group. They have true knowledge about their own intentions and the intentions of others.

The religious concept of double consciousness is a convenient way to avoid dealing with others who disagree with you. Their opposition can simply be chalked up to ignorance, because it is not possible for a member of a dominant group to know anything.

We are on dangerous ground.