Why does cathodal tDCS (negative) produce sensation, while anodal tDCS (positive) does not? If neurons fire due to an intracellular influx of positive ions, why are the negative electrodes in tDCS felt more strongly than the others? For example, see "Reducing Procedural Pain and Discomfort Associated with Transcranial Direct Current Stimulation" by McFadden et al.
Short answer
The underlying mechanism(s) causing the differences between the effectiveness of anodic and cathodic neural stimulation are largely unknown.  

Background
As far as I know, the exact difference between the effectiveness of cathodic versus anodic stimulation are largely unknown. For instance, in animals and humans, both anodic and cathodic pulses can evoke neural responses. In human cochlear implant users, the anodic phase is more effective than the cathodic phase. In contrast, in animals (guinea pigs, cats and ferrets) exactly the opposite is observed. The reason for this discrepancy is unclear (Machery &amp; Cazals, 2016).

The authors in your linked article state (McFadden, 2011): 


  We hypothesize that the mechanism behind this may be associated with a concentrated exiting of current from the scalp at the cathode.


This, honestly, makes little sense as that same current was injected at the anode, barred the electrode sizes were not different.   

Reference
- Machery &amp; Cazals, Adv Exp Med Biol (2016); 894:133-42
- McFadden et al., Brain Stimul (2011); 4(1): 38–42