Reaction-diffusion(-advection) problems are well-known in chemical engineering and computational fluid dynamics. A common feature of these systems is a linear (or non-stiff) transport sub-system and a non-linear (or stiff), highly coupled chemistry sub-system. The expected numerical effort often prohibits a fully coupled solution of the system. Therefore, the system is split into a transport and a reaction sub-system, and each sub-system is solved using specialized solvers. Operator splitting schemes are required to reconstruct the solution of the initial system from the sub-system solutions. Steady-state preserving splitting schemes are particularly essential for steady-state calculations since local time stepping (LTS) or other methods based on fictional time rely on large time steps to be efficient. This work formally analyzes common splitting schemes for reaction-diffusion problems by stability analysis and checking the steady-state preservation of a representative linear scalar problem. Balanced and Simpler Balanced splitting are the only steady-state conservative schemes analyzed. Three new steady-state preserving splitting schemes are proposed based on the findings of the formal analysis. To achieve steady-state preservation, the new schemes use splitting constants based on either the mixing derivative or the chemistry derivative. The formal analysis is accompanied by a dimensionless perfectly stirred reactor (PSR) case and a hydrogen combustion case, both known to be challenging for operator splitting schemes. The test case results are in line with the theoretical results but indicate that the scalar linear analysis is nonviable to capture the full effects of the chemical sub-system. The Simpler and the newly proposed Consistent Staggered splitting schemes give significantly better results than the remaining ones while being second-order and first-order accurate, respectively. If temporal accuracy is irrelevant, e.g., for steady-state solvers, the proposed Consistent Adaptive splitting scheme is promising since it preserves the steady-state solution first-order time accurate with less function evaluations.