  We prove the Harnack inequality for antisymmetric $s$-harmonic functions, and
more generally for solutions of fractional equations with zero-th order terms,
in a general domain. This may be used in conjunction with the method of moving
planes to obtain quantitative stability results for symmetry and overdetermined
problems for semilinear equations driven by the fractional Laplacian.
  The proof is split into two parts: an interior Harnack inequality away from
the plane of symmetry, and a boundary Harnack inequality close to the plane of
symmetry. We prove these results by first establishing the weak Harnack
inequality for super-solutions and local boundedness for sub-solutions in both
the interior and boundary case.
  En passant, we also obtain a new mean value formula for antisymmetric
$s$-harmonic functions.
