  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.
