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Well, they are fairly rigid in that the normals to their faces have to be lattice vectors. So the only deformations are to slide the faces perpendicular to themselves. (On the other hand, you may always do this, so they are not completely rigid.) And "simple" is a restriction on their shape. The image of a Delzant polytope under $GL_n(\mathbb{Z})$ is another Delzant polytope, so it only makes sense to ask about properties which are $GL_n(\mathbb{Z})$ invariant. That tends to rule out obvious geometric properties (e.g. curvature) so I'm not sure what else to say.
UPDATE: In the theme of "all related comments are welcome", here is a speculation that I haven't thought about seriously. I do know one geometric property which is preserved by $GL_n(\mathbb{R})$. For $K$ a compact convex set, let $A$ be the volume of the smallest ellipsoid containing $K$, and $a$ the volume of the largest ellipsoid contained in $K$. Let Define $s:=a/A$ to be the sphericity of $K$. (Not sure if this is the standard name.) So $s=1$ for ellipsoids, and smaller for everything else. Delzant polytopes tend to be kind of fat, so I wonder if one could prove a lower bound for their sphericity which was better than for ordinary polytopes.
Well, they are fairly rigid in that the normals to their faces have to be lattice vectors. So the only deformations are to slide the faces perpendicular to themselves. (On the other hand, you may always do this, so they are not completely rigid.) And "simple" is a restriction on their shape. The image of a Delzant polytope under $GL_n(\mathbb{Z})$ is another Delzant polytope, so it only makes sense to ask about properties which are $GL_n(\mathbb{Z})$ invariant. That tends to rule out obvious geometric properties (e.g. curvature) so I'm not sure what else to say.