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“I wouldn’t say that All Washed Up was my first choice for the title of the new record,” says Cheap Trick guitarist Rick Nielsen from the living room of his Rockford, Illinois, home. “But hey, I’ve always said that if you have to tell people you’re cool, then you’re not!”

Nielsen and his longtime bandmates, vocalist and guitarist Robin Zander and bassist Tom Petersson, are not among those who need to toot their own horns. Six decades into a career that has seen the band inducted into the Rock and Roll Hall of Fame, invited to collaborate with John Lennon, name-checked by Jerry Garcia, and cited as a primary influence by bands as disparate as Mötley Crüe and Nirvana, Cheap Trick certainly have nothing left to prove. Yet the group, whose deft ability to combine muscular riffs, edgy subject matter and unforgettable power-pop hooks continues to tour relentlessly—they’ve long since passed the 5,000-show mark—and record with a regularity that might wind bands half, or even a third, their age.