We show that the measured intrinsic octupole moments of ²²⁰ Rn, ²²⁴Ra, and ²²⁶Ra constrain the intrinsic Schiff moments of ²²⁵Ra, ²²¹Rn, ²²³Rn, ²²³Fr, ²²⁵Ra, and ²²⁹Pa. 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 noninvariant interaction produced by CP-violating fundamental physics. Those matrix elements are also correlated with octupole moments, but with a larger systematic uncertainty.