My first reaction was to think "yes", but on second thought the answer is no. Take the moment of inertia tensor around the center of mass. The moment of inertia around some axis not through the center of mass is just given by the moment around the parallel axis through the C.O.M. plus the moment due to the displacement of the C.O.M. from the new axis. That extra moment depends only on the total mass and the displacement. Thus it gives no new information about the mass distribution.
So we're back to just the inertia tensor about the C.O.M. It of course has far too little information to give the whole distribution.
In fact, in retrospect, consider the following two distributions.
A. a gram uniformly distributed on a spherical shell of radius 1 cm.
B. 1/2 gm at the middle and 1/2 gm in a spherical shell at sqrt(2)cm from the origin.
A and B have the same moments of inertia, as do an obviously infinite set of distributions from the same family.
(published on 08/14/2012)