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We consider the nonlinear problem \[(P) \;\; I u=f(x,u) \text{ in $\Omega$,}\;\; u=0 \text{ on $\mathbb{R}^{N}\setminus\Omega$ }\] in an open bounded set$\Omega\subset\mathbb{R}^{N}$, where $I$ is a nonlocal operator which may beanisotropic and may have varying order. We assume mild symmetry andmonotonicity assumptions on $I$, $\Omega$ and the nonlinearity $f$ with respectto a fixed direction, say $x_1$, and we show that any nonnegative weak solution$u$ of $(P)$ is symmetric in $x_1$. Moreover, we have the followingalternative: Either $u\equiv 0$ in $\Omega$, or $u$ is strictly decreasing in$|x_1|$. The proof relies on new maximum principles for antisymmetricsupersolutions of an associated class of linear problems.

Symmetry via antisymmetric maximum principles in nonlocal problems of variable order