  We investigate which pure states of $n$ photons in $d$ modes can be
transformed into each other via linear optics, without post-selection. In other
words, we study the local unitary (LU) equivalence classes of symmetric
many-qudit states. Writing our state as $f^\dagger|\Omega\rangle$, with
$f^\dagger$ a homogeneous polynomial in the mode creation operators, we propose
two sets of LU-invariants: (a) spectral invariants, which are the eigenvalues
of the operator $ff^\dagger$, and (b) moments, each given by the norm of the
symmetric component of a tensor power of the initial state, which can be
computed as vacuum expectation values of $f^k(f^\dagger)^k$. We provide scheme
for experimental measurement of the later, as related to the post-selection
probability of creating state $f^{\dagger k}|\Omega\rangle$ from $k$ copies of
$f^{\dagger}|\Omega\rangle$.
