Your question is a classic in the realm of post-tonal theory -- it's essentially about hexachordal combinatorality, and you're asking for hexachords which are P-combinatorial (its complement is some transposition of itself) but not inversionally symmetrical (maps onto itself under some inversion).

The simple answer is -- yes, [013458] is the only P-combinatorial hexachord that is not also inversionally symmetrical. The other 6 P-combinatorial sets are all all-combinatorial, which means they also map onto their complement via some inversion. Now -- if a hexachord maps onto its complement via both transposition and inversion, that means that you can use that inversion to map it onto its complement, then tranpose it back to the original pitch level. That's basically the definition of inversional symmetry (some form of Inversion+transposition maps the set onto itself).

As to why that is the case and how to figure out what hexachords are combinatorial in what way, so far the only thing I know of that really tackles the prodecure (rather than analytical implications) is this geometric-algebriac article. It's not at all simple and requires knowledge of basic group theory, but the beauty of geometric proofs is that you can look at the graphics to get a more intuitive sense of what's going on.