  Using the wave equation in d > or = 1 space dimensions it is illustrated how
dynamical equations may be simultaneously Poincar\'e and Galileo covariant with
respect to different sets of independent variables. This provides a method to
obtain dynamics-dependent representations of the kinematical symmetries. When
the field is a displacement function both symmetries have a physical
interpretation. For d = 1 the Lorentz structure is utilized to reveal hitherto
unnoticed features of the non-relativistic Chaplygin gas, including a
relativistic structure with a limiting case that exhibits the Carroll group,
and field-dependent symmetries and associated Noether charges. The Lorentz
transformations of the potentials naturally associated with the Chaplygin
system are given. These results prompt the search for further symmetries and it
is shown that the Chaplygin equations support a nonlinear superposition
principle. A known spacetime mixing symmetry is shown to decompose into
label-time and superposition symmetries. It is shown that a quantum mechanical
system in a stationary state behaves as a Chaplygin gas. The extension to d > 1
is used to illustrate how the physical significance of the dual symmetries is
contingent on the context by showing that Maxwells equations exhibit an exact
Galileo covariant formulation where Lorentz and gauge transformations are
represented by field-dependent symmetries. A natural conceptual and formal
framework is provided by the Lagrangian and Eulerian pictures of continuum
mechanics.
