  We prove the existence of highest, cusped, periodic travelling-wave solutions
with exact and optimal $ \alpha $-H\"older continuity in a class of fractional
negative-order dispersive equations of the form \begin{equation*} u_t + (|
\mathrm{D} |^{- \alpha} u + n(u) )_x = 0 \end{equation*} for every $ \alpha \in
(0, 1) $ with homogeneous Fourier multiplier $ | \mathrm{D} |^{ - \alpha} $. We
tackle nonlinearities $ n(u) $ of the type $ | u |^p $ or $ u | u |^{p - 1} $
for all real $ p > 1 $, and show that when $ n $ is odd, the waves also feature
antisymmetry and thus contain inverted cusps. Tools involve detailed pointwise
estimates in tandem with analytic global bifurcation, where we resolve the
issue with nonsmooth $ n $ by means of regularisation. We believe that both the
construction of highest antisymmetric waves and the regularisation of nonsmooth
terms to an analytic bifurcation setting are new in this context, with direct
applicability also to generalised versions of the Whitham, the
Burgers--Poisson, the Burgers--Hilbert, the Degasperis--Procesi, the reduced
Ostrovsky, and the bidirectional Whitham equations.
