  We present a high-field study of the strongly anisotropic easy-plane square
lattice $S$ = 2 quantum magnet Ba$_{2}$FeSi$_{2}$O$_{7}$. This compound is a
rare high-spin antiferromagnetic system with very strong easy-plane anisotropy,
such that the interplay between spin level crossings and antiferromagnetic
order can be studied. We observe a magnetic field-induced spin level crossing
occurring within an ordered state. This spin level crossing appears to preserve
the magnetic symmetry while producing a non-monotonic dependence the order
parameter magnitude. The resulting temperature-magnetic field phase diagram
exhibits two dome-shaped regions of magnetic order overlapping around 30 T. The
ground state of the lower-field dome is predominantly a linear combination of
$| S^{z} = 0 \rangle$ and $ |S^{z} = 1 \rangle$ states, while the ground state
of the higher-field dome can be approximated by a linear combination of $|
S^{z} = 1 \rangle $ and $ | S^{z} = 2\rangle$ states. At 30 T, where the spin
levels cross, the magnetization exhibits a slanted plateau, {\color {black}the
magnetocaloric effect shows a broad hump, and the electric polarization shows a
weak slope change}. We determined the detailed magnetic phase boundaries and
the spin level crossings using measurements of magnetization, electric
polarization, and the magnetocaloric effect in pulsed magnetic fields to 60 T.
We calculate these properties using a mean field theory based on direct
products of SU(5) coherent states and find good agreement. Finally, we measure
and calculate the magnetically-induced electric polarization that reflects
magnetic ordering and spin level crossings. This multiferroic behavior provides
another avenue for detecting phase boundaries and symmetry changes.
