One morning during the presidency of George H.W. Bush, Jonna Mendez, then the CIA’s chief of disguise, entered the White House wearing a mask. She had originally disguised herself as an African-American man but decided that mask wouldn’t work, not least because her voice would give her away. Instead, she borrowed the face of a female colleague. “It was a little nerve-racking,” she recalls. “I hadn’t really worn it anywhere.” She sat outside the Oval Office, chewing her pencil through the mask, until the president was ready for his morning intelligence briefing.

Entering...