The Order of the Faceless was a religious group that required its members to keep their faces hidden from others. To achieve this, they wore various objects over their faces; the example shown featured a member of the order wearing a metallic gold veil composed of a sensor web that allowed the wearer to 'see' while wearing it, although there may be other methods used to conceal the Order's faces. They apparently have no concern about seeing the faces of non-Order members, as there is no suggestion that others should cover their faces while working with a member of the Order.
In the year 2270, Qat Zaldana claimed to belong to the Order of the Faceless so she could keep her face hidden from the crew of the Federation starship USS Enterprise. (TOS novel: The Rings of Time)