Does training affect the tactile oblique effect? In the visual sciences it is known that the oblique effect can be reduced by means of training. The oblique effect is observed when testing subjects psychophysically with a grating acuity task (e.g., the BaGa test). Visual acuity is better when horizontal or vertical gratings are tested than when diagonals are used. The performance in discerning diagonal gratings has been shown to improve after training subjects, although performance in the cardinal directions stays better than the oblique stimuli. 

The oblique effect has been observed in the tactile sense too (the sense of touch). However, the amount of papers in the scientific literature is quite restricted in the tactile modality. I wasn't able to find evidence in the literature whether training can improve people's performance in tactile diagonal grating tasks. 

Can training reduce the oblique effect in the tactile modality, comparable to that observed in visual grating tasks?
Indeed, research on this particular question is somewhat lacking, but what is available suggests that training does improve performance in tactile tests of the oblique effect.

In an article published 1999, Gentaz &amp; Rossetti lament:


  Unfortunately, the effect of practice on the haptic oblique effect has
  not been yet studied.


An indication that this might soon be remedied was noted in  Junker-Tschopp, Gentaz &amp; Viviani (2010):


  We used bars instead of fuzzy stimuli (such as Gabor’s patches)
  because this study is part of a larger project in which we also tested
  learning effects in the haptic modality (blindfolded participants
  explored manually a rod with varying orientations).


It does not look like the results of this "larger project" were ever published.  However, they are reviewed in the book Psychology of Touch and Blindness by Morton A. Heller and Edouard Gentaz (one of the authors of the above mentioned papers) published in 2013 (pp 56-57):


  ... Junker-Tschopp, Gentaz, and Viviani (2010) investigated the extent
  to which visual and haptic perception are penetrable by cognitive
  factors by assessing the effect of learning on the perception of
  orientations.  The "oblique effect" questions the nature of the
  linkages between perception and cognition. ... In the haptic modality,
  the results showed that practice resulted in a general dramatic
  improvement of orientation discriminabilty.   Furthermore, the
  anisotropic bias in orientation perception is actually suppressed by a
  global learning paradigm ...


These results are in line with the idea that the persistence of the oblique effect across modalities indicates that it is situated in higher cognitive processes (it is a top-down effect).  This theory is backed by evidence demonstrating the similarity of the effect across visual and tactile modalities, so the effect of training would be expected to be similar in both.

Unfortunately, not everything is the same about the oblique effect across modalities, and authors note differences such as the role of gravity cues, memory constraints, reference frame, and while training does improve discrimination, it never eliminates the oblique effect, suggesting some bottom-up influence, and hence the potential for differences in results.

In a recent review, Mier (2014) examines the influence of training and feedback on the haptic perception of parallelity, and comes to a more guarded conclusion suggesting a more significant role for bottom-up processes:


  ... Kappers and coworkers (2008) found that the deviations were only
  marginally affected by training and feedback. Without informing their
  participants about their biased performance, they studied the effects
  of visual training (seeing the correct orientations), haptic training
  (feeling the correct orientations) and combined visuo-haptic training
  (seeing and feeling the correct orientations). In addition they
  studied the effect of error feedback on the performance of the
  participants, again under visual, haptic and visual-haptic conditions.
  They showed that the robustness of the deviations persisted even after
  participants received haptic and/or visual feedback and training.
  Haptic or visual training did not significantly decrease the magnitude
  of the deviations, only combining both training modes resulted in a
  small but significant improvement. ... Although providing participants
  with haptic and visual feedback about their errors reduced the
  deviations, performance was still far from being veridical ... As the
  authors state, it might be that extension of the feedback phase would
  eventually lead to more veridical parallel matching.


Compared to the visual realm, where training has failed to eliminate the oblique effect, the effectiveness of training on the tactile oblique effect is unclear.  The first study suggests that training is more effective on the tactile oblique effect, and is capable of entirely eliminating it, while the second study suggests that the tactile effect is less amenable to training than in the visual modality.  Note however, that the second study refers specifically to haptic perception of parallelity, which is a specific sub-case of the oblique effect where exceptionally large oblique effects have been found, and therefore may not be applicable in general.  More research is needed.