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 $f^dagger|Omegarangle$, 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 $f^{dagger k}|Omegarangle$ from $k$ copies of $f^{dagger}|Omegarangle$.
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