  Recently, the notion of topological phases of matter has been extended to
higher-order incarnations, supporting gapless modes on even lower dimensional
boundaries, such as corners and hinges. We here identify a collection of cubic
spin-3/2 fermions with biquadratic touching of Kramers degenerate valence and
conduction bands as a platform to strain-engineer higher-order topological
(HOT) phases: external uniaxial strain gives birth to a HOT Dirac semimetal or
an insulator, depending on its sign, featuring topological \emph{hinge} modes
in the strain direction. The insulator in fact exhibits \emph{mixed} topology,
and in addition supports edge modes on orthogonal planes. These outcomes are
germane when the external strain is applied along one of the $C_{4v}$ or
coordinate axes, as well as $C_{3v}$ or body-diagonal, directions. Our findings
place HgTe, gray-Sn, 227 pyrochlore iridates and half-Heusler compounds at the
frontier of strain-engineered electronic HOT phases.
