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Lamb-dips of R(0), R(2) and P(2) rovibrational transitions in the (2-0) overtone band of the HD molecule are measured at cryogenic temperatures using the technique of NICE-OHMS spectroscopy. Resulting transition frequencies are $$214\,905\,335\,240\,(100)$$ 214 905 335 240 ( 100 ) kHz for R(0), $$219\,042\,856\,794\,(150)$$ 219 042 856 794 ( 150 ) kHz for R(2) and $$206 \,898 \,802\,150 \,(150)$$ 206 898 802 150 ( 150 ) kHz for P(2). Small, but systematic deviations (at 1.7$$\sigma $$ σ level) are found from advanced ab initio calculations, supporting results from previous studies probing other lines in the (2-0) band. From a combination difference between the R(0) and P(2) lines, an accurate value for rotational-level spacing in the $$v=0$$ v = 0 ground state is determined, of $$\Delta _{(J=2)-(J=0)}= 8\,006\,533\,168\,(26)$$ Δ ( J = 2 ) - ( J = 0 ) = 8 006 533 168 ( 26 ) kHz, well in agreement with theory (at 0.6$$\sigma $$ σ ). Based on the observation of the dispersive line shape of R(0), the status of proposed models for the elusive line shapes of saturated transitions in HD is discussed.