Just a quick question - I fail at Googling this topic since I do not remember clearly, in which case (e.g. for what type of particles) is the wave function antisymmetric in terms of spatial rotation
$$ \hat{D}(\vec{o},2\pi)|\varphi\rangle ~=~ -|\varphi\rangle .$$
Any particle with half-integer spin acquires a minus sign under a $2\pi$-rotation.
The wave function of a particle with half-integer spin is multiplied by $-1$ under a $2\pi$ rotation.
The spin-statistics theorem states that a particle with integer spin obeys Bose-Einstein statistics (the equal time commutator $[,]$ vanishes) and that a particle with half-integer spin obeys Fermi-Dirac statistics (the equal time anticommutator $\{,\}$ vanishes). See e.g. Weinberg (1995) Ch. 5 for a discussion and proof. (Maybe only partial proof. See references therein.) It may be shown that the behavior of fields under rotations determines their statistics. See e.g. Cahill (2013) p. 370.
Note that a rotation by $4\pi$ leaves all wave functions invariant. I believe this is fundamentally a topological condition.
