According to Chuang's exhausted search of carbon tori with 240 atoms and simultaneously satisfying C5 rotational symmetry and IPR (Independent Pentagon Rule), there are many possible isomers for T240. Out of these compounds, I found 19 stable isomers which I discussed in our group blog before (http://byjingroup.blogspot.com/). Here is one of T240 with a shape very similar to innertube.
Moreover, I suspect this is the only fullerene with this shape. I don't have proof, though.
Optimized shape of this molecule generated by Chuang.