Enumerative questions in algebraic geometry go all the way back to the second half of the 19th century, and before; even though evolving in format and langua- ge, the main underlying idea has always been been finding a compact moduli space for the problem, and doing intersection theory on it. About twisted cu- bics, the most common moduli spaces (Hilbert schemes, stable maps, and more) partially solved all the question Schubert raised (and answered, using not enti- rely understood methods) in 1879. In this talk, we will construct a new moduli space of twisted cubics, obtained compactifying P GL4 to the so-called space of complete collineations and then taking the GIT quotient by PGL2. The space so obtained is very symmetric; in fact, following the theory of homogeneous spaces, it is possible to link plenty of geometric properties of this space to re- presentation theoretic properties of PGL4 and PGL2. In this way, intersection theory on this space becomes just a combinatorial problem involving generating functions, partition functions, and interpolation; the number 56960 of twisted cubics tangent to 12 given planes is just the integral of a piecewise polynomial over a 3 dimensional region. This is a work in progress towards my PhD thesis.