We report results and calculations of the recoil anisotropy of H and CO photofragments following laser dissociation of HCO and DCO radicals. The Ã-X̃ transition of HCO/DCO is known to lie perpendicular to the molecular plane. Excitation using linearly polarized light, however, gives rise to fragments recoiling parallel to the axis of polarization. The degree of spatial anisotropy was found to depend as expected on the value of K (the projection of J onto the a axis) for the upper electronic state, but also to depend on the initial value of K in the lower state, contrary to normal expectation. The anisotropy parameter, β, was measured for the following K′ ← K″ transitions: 1 ← 0, β = 1.0 ± 0.1; 1 ← 2, β = 0.0 ± 0.15; 0 ← 1, Q branch, β = 0.25 ± 0.15; 0 ← 1, R branch, β = 0.0 ± 0.15; and 2 ← 1, β = 0.1 ± 0.2 These unusual results are explained by examining the detailed mechanism of the Renner-Teller interaction and by calculating the angle of recoil of the H atom with respect to the CO molecule.