Many of the families owned businesses that clashed with environmental or workplace regulators, come under federal or state investigation, or waged battles over their tax bills with the Internal Revenue Service, Ms. Mayer reports. The Kochs’ vast political network, a major force in Republican politics today, was “originally designed as a means of off-loading the costs of the Koch Industries environmental and regulatory fights onto others” by persuading other rich business owners to contribute to Koch-controlled political groups, Ms. Mayer writes, citing an associate of the two brothers.

Mr. Scaife, who died in 2014, donated upward of a billion dollars to conservative causes, according to “Dark Money,” which cites his own unpublished memoirs. Mr. Scaife was driven in part, Ms. Mayer writes, by a tax loophole that granted him his inheritance tax free through a trust, so long as the trust donated its net income to charity for 20 years. “Isn’t it grand how tax law gets written?” Mr. Scaife wrote.

In Ms. Mayer’s telling, the Kochs helped bankroll — through a skein of nonprofit organizations with minimal public disclosure — decades of victories in state capitals and in Washington, often leaving no fingerprints. She credits groups financed by the Kochs and their allies with providing support for the Tea Party movement, along with the public relations strategies used to shrink public support for the Affordable Care Act and for President Obama’s proposals to mitigate climate change.

The Koch network also provided funding to fine-tune budget proposals from Representative Paul D. Ryan, such as cuts to Social Security, so they would be more palatable to voters, according to the book. The Kochs were so influential among conservative lawmakers, Ms. Mayer reports, that in 2011, Representative John A. Boehner, then the House speaker, visited David Koch to ask for his help in resolving a debt ceiling stalemate.

“Dark Money” also contains revelations from a private history of the Kochs commissioned by David’s twin brother, William, during a lengthy legal battle with Charles and David over control of Koch Industries.

Ms. Mayer describes a sealed 1982 deposition in which William Koch recalled participating in an attempt by Charles and David to blackmail their fourth and eldest brother, Frederick, into relinquishing any claim to the family business by threatening to tell their father that he was gay.

David Koch has since described himself as socially liberal and as a supporter of same-sex marriage.