It is shown that a two-level orthogonal array with strength three whose run size is not a multiple of 16 has the hidden projection property that in its projection on to any five factors, all the main effects and two-factor interactions are estimable if the hgher-order interactions are neagible. The result applies to the fold-over of any strength-two orthogonal array whose run size is not a multiple of 8. This extends Diamond's (1995) result on the fold-over of the 12-run Plackett-Burman design. Some connections with search designs are also presented.