What is the neural adaptation underlying the habituation to reversed visual images? When a person wears glasses that reverse the image upside-down, it is possible to adjust your vision in a matter of days. What are the suggestions behind the neurological process?
Short answer
From what I know from the literature, adapting to prism glasses mainly finds it origins in adaptation of behavioral motor control.

Background
The adaptation to wearing prism glasses is often ascribed to the adaptation of the proprioceptive system, i.e., subjects adapt their motor behavior to compensate for the visual field shift (e.g., Richter et al. (2002), Hatada &amp; Rosetti (2006)), or through shifts in their egocentric reference frame. Admittedly, some subtle, short-lasting after-effects do seem to occur in the visual domain (Hatada et al., 2006), but gross adaptation is mainly centered on behavioral adaptations of motor control, at least as far as I am aware. 

In four human subjects that wore prism glasses for about 36 days it was shown that after a week or two, also their ipsilateral visual cortices were activated, while normally only the contralateral cortex is activated (Miyauchi et al., 2004). So there is a definite visuo-neurophysiological correlate of wearing prism glasses. Nonetheless The authors conclude that... 


  [The] changes observed [in] adaptation to left–right reversing may be associated with the process of [...] strategic perceptual-motor control. [...]. For example, when the subjects wearing left–right reversing goggles pick up an object located in the left visual field, they have to use their right hand which is normally located in the right visual field [...].   


Note that the link you provide describes mainly anecdotal reports from a single researcher. Above studies included multiple subjects. 

References
- Hatada et al., Exp Brain Res (2006); 173(3): 415–24
- Hatada et al., Exp Brain Res (2006); 174(1): 189–98
- Hatada &amp; Rosetti, Exp Brain Res (2006); 169(3): 417–26
- Miyauchi et al., J Physiol - Paris (2004); 98: 207–19
- Richter et al., Exp Brain Res (2002); 144(4): 445–57