Calorimetric and magnetic measurements of AC loss have been performed on stabrite-coated Rutherford cable conforming to the Large Hadron Collider (LHC) winding (28 strand) specification, with the field directed either perpendicular to the broad face (the face-on, FO, orientation) or parallel to it (edge-on, EO). It was found that the insertion of a thin metallic or insulating core into the cable suppressed the FO AC loss under typical conditions by a factor of 10 and rendered it practically insensitive to the application of cold uniaxial stress (of up to about 78 MPa). The FO loss having been suppressed, it could also be controlled (i.e. finely adjusted) by changing the level compaction (by varying core thickness at thickness) or external compaction (by changing the overall thickness of the cable). Of course the EO loss was much less sensitive to the presence of a core, the use of which therefore enabled the FO and EO losses to be independently adjusted.