You absolutely can, and there's even been recent research into the computational mechanisms that are behind this kind of inference [1]. They build a computational model that learns spatial/visual representation from (simulated) haptic stimuli with performance similar to human subjects.
I think the reason Sinha's experiment had a negative result was, as other commenters have mentioned, that this ability is a learned behavior. Those who are blind from birth simply never developed the ability to form abstract representations from visual stimuli, therefore haptic information cannot be translated into these visual representations. The other commenters who are saying there is no pathway between the two modalities are wrong, and it's easy to set up an experiment to show it. Just blindfold someone and hand them an unfamiliar object, then, without showing it to them, have them draw it. It's clear that we can do this to some extent.
Right, there is a physical difference between the brain of a person who has been blind from birth and someone who has not. This also means that if you restore sight to someone who once had it and then lost it, they should have no trouble picking out an object they hadn't seen (only felt), even if it was an object they had neither seen nor felt prior to losing their vision.
I think the reason Sinha's experiment had a negative result was, as other commenters have mentioned, that this ability is a learned behavior. Those who are blind from birth simply never developed the ability to form abstract representations from visual stimuli, therefore haptic information cannot be translated into these visual representations. The other commenters who are saying there is no pathway between the two modalities are wrong, and it's easy to set up an experiment to show it. Just blindfold someone and hand them an unfamiliar object, then, without showing it to them, have them draw it. It's clear that we can do this to some extent.
[1] http://www.bcs.rochester.edu/people/robbie/jacobslab/abstrac...