  We discuss the classification problem for the unitary easy quantum groups,
under strong axioms, of noncommutative geometric nature. Our main results
concern the intermediate easy quantum groups $O_N\subset G\subset U_N^+$. To
any such quantum group we associate its Schur-Weyl twist $\bar{G}$, two
noncommutative spheres $S,\bar{S}$, a noncommutative torus $T$, and a quantum
reflection group $K$. Studying $(S,\bar{S},T,K,G,\bar{G})$ leads then to some
natural axioms, which can be used in order to investigate $G$ itself. We prove
that the main examples are covered by our formalism, and we conjecture that in
what concerns the case $U_N\subset G\subset U_N^+$, our axioms should restrict
the list of known examples.
