An assembly such as an anchor screw, bone plate, offset hook, post, transverse connector or other spinal anchor for anchoring to bone and clamping a linkage such as a rod, wire cable or the like. The assembly has a top member with an open slot to receive the linkage, and a twist-lock closure cap to close the open end, capturing the linkage in the slot. One closure cap fits over and around the top member, with a set of segmented protrusions that extend through a corresponding set of protruding flange segments spaced about the circumference of the top member. The cap is configured to rotate and lock against the top member like a flange-locking bayonet mount when turned through a limited degree of rotation. Slots may be provided through the face of the closure cap allowing reduction tabs to project through the cap, and the locking cap may be adapted to a range of screw, eye, hook, post, connector and other anchor assemblies to securely clamp the linkage by use of a rotation tool disposed along an access path aligned with the axis of the screw, thus requiring no lateral clearance along the longitudinal direction of the linkage element. Locking is effected by a set screw that passes through the cap and tightens against the linkage to firmly immobilize it in the anchor assembly. Another cap embodiment fits into the center of the head with a partial twist locking motion. Opposed locking surfaces prevent slippage in the radial direction, and a sloping contact floor surface may apply torque to further increase pressure on urge mating detent features in an adjacent wall.