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

Which multiphoton states are related via linear optics?