  Using first principles density functional theory calculations, we show how
Rashba-type energy band splitting in the hybrid organic-inorganic halide
perovskites APbX$_3$ (A=CH$_3$NH$_3^+$, CH(NH$_2$)$_2^+$, Cs$^+$ and X=I, Br)
can be tuned and enhanced with electric fields and anisotropic strain. In
particular, we demonstrate that the magnitude of the Rashba splitting of
tetragonal (CH$_3$NH$_3$)PbI$_3$ grows with increasing macroscopic alignment of
the organic cations and electric polarization, indicating appreciable
tunability with experimentally-feasible applied fields, even at room
temperature. Further, we quantify the degree to which this effect can be tuned
via chemical substitution at the A and X sites, which alters amplitudes of
different polar distortion patterns of the inorganic PbX$_3$ cage that directly
impact Rashba splitting. In addition, we predict that polar phases of CsPbI$_3$
and (CH$_3$NH$_3$)PbI$_3$ with $R3c$ symmetry possessing considerable Rashba
splitting might be accessible at room temperature via anisotropic strain
induced by epitaxy, even in the absence of electric fields.
