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The complexity of the problem of deciding properties expressible in FO logic on graphs -- the FO model checking problem (parameterized by the respective FO formula), is well-understood on so-called sparse graph classes, but much less understood on hereditary dense graph classes. Regarding the latter, a recent concept of twin-width [Bonnet et al., FOCS 2020] appears to be very useful. For instance, the question of these authors [CGTA 2019] about where is the exact limit of fixed-parameter tractability of FO model checking on permutation graphs has been answered by Bonnet et al. in 2020 quite easily, using the newly introduced twin-width. We prove that such exact characterization of hereditary subclasses with tractable FO model checking naturally extends from permutation to circle graphs (the intersection graphs of chords in a circle). Namely, we prove that under usual complexity assumptions, FO model checking of a hereditary class of circle graphs is in FPT if and only if the class excludes some permutation graph. We also prove a similar excluded-subgraphs characterization for hereditary classes of interval graphs with FO model checking in FPT, which concludes the line a research of interval classes with tractable FO model checking started in [Ganian et al., ICALP 2013]. The mathematical side of the presented characterizations -- about when subclasses of the classes of circle and permutation graphs have bounded twin-width, moreover extends to so-called bounded perturbations of these classes.