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MICHOUD, La.—Look at NASA's high-profile Orion spacecraft, and you may get a funny feeling of familiarity. While the modern crew vehicle recently made its big screen debut in the Oscar-nominated The Martian, any lingering deja vu more likely comes from a different place. With the Orion module, there's more than a passing resemblance to its predecessor—the one from the Apollo program.

"To the untrained eye, it looks very much the same," says...