This society is unusually well suited to showing where a reliable account stops.
Documented: it was founded at Yale in 1832; it selects a small fixed number of undergraduates each year, fifteen in the modern pattern; it owns a windowless building on campus known as the Tomb and a private island retreat; it began admitting women in 1992 after a dispute among its own graduates; and its membership rolls, published periodically over its history, include two presidents named Bush, a chief justice, cabinet secretaries, senators, and a large number of figures in finance, publishing and intelligence. The membership is not secret and never has been.
Reported on the evidence of members, and credible without being verifiable: initiation involves confessional autobiography before the group, and the initiation ceremony makes use of the building's collection of funerary objects. Alexandra Robbins's account, based on interviews with a number of members, is the fullest treatment and is where most of this comes from. It should be read as journalism resting on informants rather than as documentary history.
Not documented, and asserted mainly by writers with a general thesis to advance: that the society directs American policy, that it constitutes a mechanism of dynastic control, or that it is a component of any larger structure. What the evidence supports is narrower and duller. A society drawing from a small number of Yale undergraduates for nearly two centuries produced members who went on to prominence, which is what selective recruitment from an elite institution produces regardless of what the society does with its evenings. Membership overlaps heavily with the American governing class because it was drawn from it.
The building's contents are the one place where a specific claim is worth stating carefully. It has long been alleged that human remains held there include the skull of Geronimo, taken from his grave at Fort Sill in 1918. Descendants brought suit in 2009 seeking their return. The case was dismissed on procedural grounds without any determination of what the society holds, and the question is therefore open rather than settled in either direction.