  Superconductivity in a crystalline lattice without inversion is subject to
complex spin-orbit-coupling effects, which can lead to mixed-parity pairing and
an unusual magnetic response. In this study, the properties of a layered
superconductor with alternating Rashba spin-orbit coupling in the stacking of
layers, hence (globally) possessing a center of inversion, is analyzed in an
applied magnetic field, using a generalized Ginzburg-Landau model. The
superconducting order parameter consists of an even- and an odd-parity pairing
component which exchange their roles as dominant pairing channel upon
increasing the magnetic field. This leads to an unusual kink feature in the
upper critical field and a first-order phase transition within the mixed phase.
We investigate various signatures of this internal phase transition. The
physics we discuss here could explain the recently found $H$--$T$ phase diagram
of the heavy Fermion superconductor CeRh$_2$As$_2$.
