jats:titleAbstract</jats:title> jats:pIn this work we introduce the jats:italicr</jats:italic>-Hunter–Saxton equation, a generalisation of the Hunter–Saxton equation arising as extremals of an action principle posed in jats:italicL</jats:italic> jats:sup jats:italicr</jats:italic> </jats:sup>. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the jats:italicr</jats:italic>-Hunter–Saxton equation.</jats:p>