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We propose a novel method to align accelerating structures such as those used in the Compact Linear Collider (CLIC) by exploiting a mode that copropagates with the normal accelerating mode. This mode has an octupolar dependence in the transverse direction and is caused by radial waveguides intended to damp higher-order modes. The nonlinear dependence of the octupolar mode makes it possible to determine the center of the structure from the nonlinear dependence of the transverse kick, observed on a downstream beam position monitor, while changing the transverse position of the beam with respect to the accelerating structures. We discuss the method, its tolerances and disentangling the individual misalignments of two adjacent accelerating structures that are powered from a single source.