Previous studies [Perception 28 (1999) 1001; Perception 28 (1999) 781] on the haptic perception of parallelity on a horizontal plane showed that what subjects haptically perceive as being parallel deviates considerably from what is physically parallel. The deviations could be described with a subject-dependent orientation gradient in the left-right direction. The gradients found in the bimanual conditions were significantly larger (about 70%) than those in the unimanual conditions. The questions to be answered in the present study are the following: (1) Does the haptic perception of parallelity in the midsagittal plane also show systematic deviations from veridicality? (2) Are the unimanual and bimanual performances again quantitatively but not qualitatively different? The set-up consisted of a plate positioned in the midsagittal plane of the subject. The subject touched the right side of the plate with his/her right hand and the left side with the left hand. The results show again large systematic deviations. The major part of the deviations can be described by means of a subject-dependent orientation gradient in the vertical direction. The quantitative (but not qualitative) difference between the unimanual and the bimanual conditions is much larger in the midsagittal plane than in the horizontal plane.