We use symmetry analysis, density functional theory calculations, and k·p modeling to scrutinize Rashba and Dresselhaus effects in hybrid organic-inorganic halide perovskites. These perovskites are at the center of a recent revolution in the field of photovoltaics but have also demonstrated potential for optoelectronic applications such as transistors and light emitters. Due to a large spin-orbit coupling of the most frequently used metals, they are also predicted to offer a promising avenue for spin-based applications. With an in-depth inspection of the electronic structures and bulk lattice symmetries of a variety of systems, we analyze the origin of the spin splitting in two- and three-dimensional hybrid perovskites. It is shown that low-dimensional nanostructures made of CH3NH3PbX3 (X = I, Br) lead to spin splittings that can be controlled by an applied electric field. These findings further open the door for a perovskite-based spintronics.