We show that the measured intrinsic octupole moments of \(^{220}\)Rn, \(^{224}\)Ra, and \(^{226}\)Ra constrain the intrinsic Schiff moments of \(^{225}\)Ra, \(^{221}\)Rn, \(^{223}\)Rn, \(^{223}\)Fr, \(^{225}\)Ra, and \(^{229}\)Pa. The result is a dramatically reduced uncertainty in intrinsic Schiff moments. Direct measurements of octupole moments in odd nuclei will reduce the uncertainty even more. The only significant source of nuclear-physics error in the laboratory Schiff moments will then be the intrinsic matrix elements of the time-reversal non-invariant interaction produced by CP-violating fundamental physics. Those matrix elements are also correlated with octupole moments, but with a larger systematic uncertainty.