We study the development of caustics in shift-symmetric scalar field theories by focusing on simple waves with an
S
O
(
p
)
-symmetry in an arbitrary number of space dimensions. We show that the pure Galileon, the DBI–Galileon, and the extreme-relativistic Galileon naturally emerge as the unique set of caustic-free theories, highlighting a link between the caustic-free condition for simple
S
O
(
p
)
-waves and the existence of either a global Galilean symmetry or a global (extreme-)relativistic Galilean symmetry.