  We report the discovery of two-phase unconventional superconductivity in
CeRh$_2$As$_2$. Using thermodynamic probes, we establish that the
superconducting critical field of its high-field phase is as high as 14 T,
remarkable in a material whose transition temperature is 0.26 K. Furthermore, a
$c$-axis field drives a transition between two different superconducting
phases. In spite of the fact that CeRh$_2$As$_2$ is globally centrosymmetric,
we show that local inversion-symmetry breaking at the Ce sites enables Rashba
spin-orbit coupling to play a key role in the underlying physics. More detailed
analysis identifies the transition from the low- to high-field states to be
associated with one between states of even and odd parity.
