Effective field theory methods suggest that some rather-general extensions of General Relativity include, or are mimicked by, certain higher-order curvature corrections, with coupling constants expected to be small but otherwise arbitrary. Thus, the tantalizing prospect to test the fundamental nature of gravity with gravitational-wave observations, in a systematic way, emerges naturally. Here, we build black hole solutions in such a framework and study their main properties. Once rotation is included, we find the first purely gravitational example of geometries without $mathbb{Z}_2$-symmetry. Despite the higher-order operators of the theory, we show that linearized fluctuations of such geometries obey second-order differential equations. We find nonzero tidal Love numbers. We study and compute the quasinormal modes of such geometries. These results are of interest to gravitational-wave science but also potentially relevant for electromagnetic observations of the galactic center or $X$-ray binaries.
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