President Trump said he “hereby ordered” U.S. companies doing business in China to explore relocating their operations and stiffened tariffs on Chinese imports after Beijing unveiled its own new levies on American goods, the latest twists in a trade war rattling investors and confounding central bankers.

Mr. Trump tweeted late Friday afternoon he would raise the tariff rate on existing and planned tariffs by 5 percentage points. Tariffs already in place on about $250 billion of Chinese goods will rise to 30% Oct. 1. Tariffs...