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Recently it has been argued that near-horizon modifications of the standard(classical) black hole spacetime could lead to observable alterations of thegravitational waveform generated by a binary black hole coalescence. Suchmodifications can be inspired by quantum gravity considerations, resulting inspeculative horizonless exotic compact objects (ECOs) with no singularities,which may be an alternative to the classical black hole solution. A largelymodel-independent description of these objects proposed in the literaturerelies on the introduction of a partially reflective wall at some smalldistance away from the "would-be" horizon. The inspiral-merger-ringdown of apair of such objects would be subject to possibly detectable deviations fromthe black hole case due to matter effects. In particular, the ringdown phasewould be modified by the late emergence of so-called "echoes" in the waveform,but most studies so far have considered spherically symmetric backgrounds. Weuse an in-falling scalar charge as an initial perturbation to simulate theexcitation of the echoes of a rotating ECO and we explore both the co-rotatingand counter-rotating cases, which provide distinct signals. In particular,rotation breaks the symmetry between positive and negative frequencies andintroduces a subdominant frequency contribution in each echo, which we examinehere for the first time. Our results follow consistently from the solution ofthe Teukolsky equation using the MST method developed by Mano, Suzuki andTakasugi, and the construction of the complex Green's function integrated overdifferent particle geodesics.

Spicing up the recipe for echoes from exotic compact objects: orbital differences and corrections in rotating backgrounds