We use a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms, in combination with the method of moving planes, to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that non-negative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set
Ω
⊂
R
n
\Omega \subset \mathbb {R}^n
must be radially symmetric if one of their level surfaces is parallel to the boundary of
Ω
\Omega
; in turn,
Ω
\Omega
must be a ball.
Furthermore, we discuss maximum principles and the Harnack inequality for antisymmetric functions in the fractional setting and provide counter-examples to these theorems when only ‘local’ assumptions are imposed on the solutions. The construction of these counter-examples relies on an approximation result that states that ‘all antisymmetric functions are locally antisymmetric and
s
s
-harmonic up to a small error’.