What is Exchange?

For two identical particles with state space spanned by products $\ket{a}\otimes\ket{b}$ (particle 1 in $\ket a$, particle 2 in $\ket b$ — positions, spins and all), the exchange operator $\hat P_{12}$ swaps the roles: $\hat P_{12}\ket a\otimes\ket b = \ket b\otimes\ket a$. It is Hermitian, squares to the identity, and commutes with any Hamiltonian that treats the particles identically: its eigenvalues $\pm1$ are conserved labels of the world.