Adaptive-resolution particle methods reduce the computational cost for problems that develop a wide spectrum of length scales in their solution. Concepts from self-organization can be used to determine suitable particle distributions, sizes, and numbers at runtime. If the spatial derivatives of the function strongly depend on the direction, the computational cost and the required number of particles can be further reduced by using anisotropic particles. Anisotropic particles have ellipsoidal influence regions (shapes) that are locally aligned with the direction of smallest variation of the function. We present a framework that allows consistent evaluation of linear differential operators on arbitrary distributions of anisotropic particles. We further extend the concept of particle self-organization to anisotropic particles, where also the directions and magnitudes of anisotropy are self-adapted. We benchmark the accuracy and efficiency of the method in a number of 2D and 3D test cases.