Is it possible for certain people to perceive colors differently? What if someone perceives a color as 'red' when it is actually 'green'?

Since different people have preferences for different colors, and colors are perhaps constructed in the mind, is it possible that the entire space of colors can be experienced quite differently for some people?
Yes, this scenario is possible, occurring with certain cases of brain lesions in specific areas of the visual cortex, the fusiform, lingual and posterior parahippocampal gyri.  These areas are analogous to what is referred to in primates as V4, or the 4th visual cortex, and are known to be involved (at least partly) in the perception of color (though see this commentary).


 Courtesy wikipedia 

In this study, subjects with brain lesions were asked to identify colors in a blue ambient light (meant to mimic the outdoors) and under a reddish light (meant to mimic indoor lighting).  


  ... the three patients...with abnormal matches on the blue, green and red chips made different matches than control subjects for pink, violet, turquoise and green targets... Their matches for these chips were shifted towards orange-red for the pink and violet
  targets and towards green for the turqoise target. One patient displayed a shift toward turquois[e] for the green target.


So, the level at which this "signal" is getting crossed can only be inferred from the level of the lesion, but presumably, the transduction of color by the cones and the processing of the colors in the P and K cells of the Lateral Geniculate is all still intact.
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  Clarke, S., Walsh, V.,Schoppig, A.,Assal, G., Cowey, A. (1998) Colour constancy impairments in patients with lesions of the prestriate cortex. Exp Brain Res 123:154-158.