Quantum theory stipulates that if two particles are identical in all physical aspects, the allowed states describing the system are either symmetric or antisymmetric with respect to permutations of single-particle states. Experimentally, the symmetry of the states can be inferred indirectly from the fact that neglecting the correct exchange symmetry in the theoretical analysis leads to dramatic discrepancies with the observations. The only way to directly unveil the symmetry of the states for, say, two identical particles is through the interference of the state itself and its physically permuted version, and measuring the phase associated with the permutation process, the so-called particle exchange phase. Following this idea, we have observed the exchange phase of indistinguishable photons, providing direct evidence of their bosonic character.