  For any $n\geq 3$, we prove that there exist equivalences between these
apparently unrelated objects: irreducible $n$-dimensional non degenerate
projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal
scrolls and 3-covered by twisted cubic curves, up to projective equivalence;
quadro-quadric Cremona transformations of $ \mathbb P^{n-1}$, up to linear
equivalence; $n$-dimensional complex Jordan algebras of rank three, up to
isotopy.
  We also provide some applications to the classification of particular classes
of varieties in the class defined above and of quadro-quadric Cremona
transformations, proving also a structure theorem for these birational maps and
for varieties 3-covered by twisted cubics by reinterpreting for these objects
the solvability of the radical of a Jordan algebra.
