The high-resolution laser-induced fluorescence spectra of the Ne.OH (A) over tilde (0, 1(1), 1) <-- (X) over tilde (0, 0(0), 0) and Ne.OD (A) over tilde (0, 0(0), 0) <-- (X) over tilde (0, 0(0), 0), (A) over tilde (0, 0(0), 1) <-- (X) over tilde (0, 0(0), 0), (A) over tilde (0, 1(0), 1) <-- (X) over tilde (0, 0(0), 0), (A) over tilde (0, 1(1), 1) <-- (X) over tilde (0, 0(0), 0), transitions were obtained. Analyses of these spectra determine precise bond lengths and reveal interesting spin-rotation and parity structure. In the (A) over tilde (0, 1(K=0,1),1) levels of Ne.OH and Ne.OD, the spin-rotation structure is described with the parameter gamma, whose magnitude is larger than in the other observed levels of Ne.OH, Ne.OD, and Ar.OH, and Ar.OD. For the K = 1 level an additional term kappa is introduced and is found to be an order of magnitude larger than gamma. A model based on perturbation theory is proposed and successfully explains the peculiar spin-rotation structure in the (A) over tilde (0, 1(K), 1) levels. In addition, a parity-splitting (P-type doubling) for the (X) over tilde (0, 0(0), 0) level of Ne.OD was determined to be a factor of 10 larger than that in Ar.OD. The parity splitting (K-type doubling) for the (A) over tilde (0, 1(1), 1) level of Ne.OH is also observed and measured.