I was reading this and it mentions in the 3-electron section, that for a spacial wave function to be symmetric under fermion swapping, it must be a function of even parity. Similarly for anti-symmetry under fermion swapping, it must be a function of odd parity.
It is not immediately obvious to me why parity symmetry $(-1)^l$ has anything to do with the bosonic or fermionic properties of the spacial wave function. So I suppose my questions are:
Is it true that spacial inversion is the same symmetry as swapping?
And if so, why?