The rotational excitation of a singly deuterated water molecule (HDO) by a heavy atom (Ne) and a light diatomic molecule (H2) is investigated theoretically and experimentally in the near-threshold regime. Crossed-molecular-beam measurements with a variable crossing angle are compared to close-coupling calculations based on high-accuracy potential energy surfaces. The two lowest rotational transitions, 000 → 101 and 000 → 111, are probed in detail and a good agreement between theory and experiment is observed for both transitions in the case of HDO + Ne, where scattering resonances are however blurred out experimentally. In the case of HDO + H2, the predicted theoretical overlapping resonances are faithfully reproduced by experiment for the 000 → 111 transition, while the calculated strong signal for the 000 → 101 transition is not detected. Future work is needed to reconcile this discrepancy.